LOG-CANONICAL POISSON STRUCTURES AND NON-COMMUTATIVE INTEGRABLE SYSTEMS By Nicholas Ovenhouse A DISSERTATION Michigan State University in partial ful(cid:27)llment of the requirements Submitted to for the degree of Mathematics — Doctor of Philosophy 2019 ABSTRACT LOG-CANONICAL POISSON STRUCTURES AND NON-COMMUTATIVE INTEGRABLE SYSTEMS By Nicholas Ovenhouse Log-canonical Poisson structures are a particularly simple type of bracket which are given by quadratic expressions in local coordinates. They appear in many places, including the study of cluster algebras. A Poisson bracket is “compatible” with a cluster algebra structure if the bracket is log-canonical with respect to each cluster. In joint work with John Machacek, we prove a structural result about such Poisson structures, which justi(cid:27)es the use and signi(cid:27)cance of such brackets in cluster theory. The result says that no rational coordinate-changes can transform these brackets into a simpler form. The pentagram map is a discrete dynamical system on the space of plane polygons (cid:27)rst intro- duced by Schwartz in 1992. It was proved to be Liouville integrable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to certain networks embedded in a disc or annulus, and its relation to cluster algebras. These Poisson structures are log-canonical. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram map in terms of these networks, and used the associated Poisson structures to give a new proof of integrability. In 2011, Mari Be(cid:29)a and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it had a Lax representation. We reinterpret this Grassmann penta- gram map in terms of non-commutative algebra, in particular the double brackets of Van den Bergh, and generalize the approach of Gekhtman et al. to establish a non-commutative version of integrability. The integrability of the pentagram maps in both projective space and the Grass- mannian follow from this more general algebraic system by projecting to representation spaces. ACKNOWLEDGMENTS I would like to thank all of the faculty in the mathematics department whose classes I have at- tended over the years. I also thank my fellow graduate students for their friendship and support — especially Andrew Claussen, Du(cid:29) Baker-Jarvis, John Machacek, and Blake Icabone. Thanks also to the members of my dissertation and comprehensive exam committees — Misha Shapiro, Bruce Sagan, Rajesh Kulkarni, Jon Hall, and Linhui Shen. Thanks to Semeon Artamonov and Leonid Chekhov for helpful discussions that contributed to the work presented in this dissertation. I would like to give special thanks to Tianran Chen and T.Y. Li for hiring me as a research assistant when I was an undergraduate. It was this experience of working with Tianran that (cid:27)rst inspired me to pursue a graduate degree in mathematics. A very special thanks to my advisor, Misha Shapiro. His presence as a teacher, mentor, col- league, and friend have helped to make my experience in graduate school a very positive and rewarding one. I am very grateful for all he has taught me, both about mathematics itself and the profession of being a research mathematician. Finally, I would like to thank my family for their love and support — especially my parents, my brother Alex, and my wife Kristy. iv TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Poisson Structures . 1.1 . 1.2 Classes of Poisson Structures . 1.3 Algebraically Reduced Brackets Log-Canonical Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 The Pentagram Map . . . . . . 2.1 Background . . 2.2 Twisted Polygons and the Pentagram Map . . 2.3 Corrugated Polygons and Higher Pentagram Maps . 2.4 Coordinates on the Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . viii . . . . 1 1 2 4 . . . . . 8 8 9 11 12 . 15 15 . . 20 . 28 . 28 29 . 32 . . 34 . 38 38 . . 40 41 . . 51 . 54 54 . . 59 59 . 60 . 62 . . 64 67 . . 75 Chapter 3 Networks and Poisson Structures . . . . . Poisson Structures on the Space of Edge Weights . 3.1 Weighted Directed Fat Graphs . 3.2 . . . . . . . . . . . . Chapter 4 . 4.1 The Quiver and Poisson Bracket . . . 4.2 The x, y Coordinates 4.3 The p, q Coordinates . . . . 4.4 The Postnikov Moves and the Invariants Interpreting the Pentagram Map in Terms of Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5 The Grassmann Pentagram Map . . . 5.1 Twisted Grassmann Polygons and the Pentagram Map . . 5.2 Corrugated Grassmann Polygons and Higher Pentagram Maps . . 5.3 Description of the Moduli Space by Networks 5.4 Expression for the Pentagram Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 6 6.1 Double Brackets . Non-Commutative Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 7 . 7.1 Weighted Directed Fat Graphs (Revisited) . . . 7.2 The Free Skew Field and Mal’cev Neumann Series . . . 7.3 An Example of a Series Expansion . . . 7.4 Double Brackets Associated to a Quiver . . 7.5 A Formula for the Bracket . . . 7.6 Goldman’s Bracket and the Twisted Ribbon Surface . Non-Commutative Networks and Double Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 The X, Y Variables 7.8 The P, Q Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8 Invariance, Invariants, and Integrability . . . . . . . Invariance of the Induced Bracket . . . . . 8.1 8.2 The Invariants . 8.3 . Involutivity of the Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 9 Chapter 10 REFERENCES . Recovering the Lax Representation . Conclusions and Further Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 82 . 83 . 83 87 . . 89 . 95 . 99 . 100 vi LIST OF TABLES Table 3.1: De(cid:27)nition of ε(f, g) for parallel paths . . . Table 7.1: Contributions to {{f, g}} from Figure 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 72 vii LIST OF FIGURES Figure 2.1: The pentagram map for a hexagon . . . Figure 3.1: Types of vertices in a perfect network . Figure 3.2: Local Postnikov moves . . . . . . Figure 3.3: Concatenation of local pictures . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.4: Two parallel paths sharing a common subpath . Figure 4.1: Figure 4.2: Figure 4.3: . . . . . . . . . . . . . . . The network Q3,5 . The quiver Q3,5 after gauge transformations . The weights xi, yi, z represented as cycles . . The dual quiver Q∨ . . . . . . Figure 4.4: . . Figure 5.1: A generator of the gauge subgroup G◦ . . . Figure 7.1: Non-commutative Postnikov moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 16 17 21 25 29 31 32 33 43 60 72 84 92 92 97 Figure 7.2: Orientations of the beginning of a common subpath . Figure 8.1: Edge weights after white-swap and black-swap moves . Figure 8.2: A “type I” swap . Figure 8.3: A “type II” swap . . . . . . . Figure 9.1: Elementary Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Chapter 1 Log-Canonical Poisson Structures 1.1 Poisson Structures In this section we give some basic de(cid:27)nitions and background on Poisson geometry. All results in this section are well-known and can be found in any introductory text (e.g. [LGPV12]). De(cid:27)nition 1. A vector space A is a Poisson algebra if • (A,·) is an associative, commutative algebra • (A,{−,−}) is a Lie algebra • The two operations satisfy the Leibniz identity: {a, b · c} = {a, b} · c + b · {a, c} De(cid:27)nition 2. A manifold M is called a Poisson manifold if its algebra of functions C∞(M ) is a Poisson algebra. Similarly, an algebraic variety X is called a Poisson variety if its ring of regular functions OX is a Poisson algebra. Example. For any (cid:27)eld K, the polynomial algebra K[x1, y1, . . . , xn, yn] (with its usual multi- plication) is a Poisson algebra with the bracket given by {xi, yj} = δij, and all other brackets between generators being zero. This makes the a(cid:28)ne space K2n into a Poisson variety. 1 Example. Suppose (M, ω) is a symplectic manifold. Then the symplectic form ω naturally in- duces a Poisson structure on M in the following way. Since ω is non-degenerate, it induces an isomorphism T M ∼= T∗M via v (cid:55)→ ω(v,−). The Leibniz rule implies that for each f ∈ C∞(M ), the map {f,−} is a derivation, and hence de(cid:27)nes a vector (cid:27)eld. We denote this vector (cid:27)eld by Xf, and call it the Hamiltonian vector (cid:27)eld associated to f. We may then de(cid:27)ne a Poisson bracket on C∞(M ) given by {f, g} = ω(Xf , Xg) Skew-symmetry of the bracket follows immediately from the skew-symmetry of ω. The Jacobi identity is equivalent to the fact that ω is a closed form (i.e. dω = 0). are the structure constants of g in this basis, so that [xi, xj] =(cid:80) Example. Let g be a (cid:27)nite-dimensional Lie algebra, of dimension n. Suppose that x1, . . . , xn is a ijxk. basis of g. Suppose ck ij The symmetric algebra of g is isomorphic to the polynomial ring K[x1, . . . , xn]. We may de(cid:27)ne a Poisson bracket on the polynomial ring using these structure constants: k ck {xi, xj} = (cid:88) k ck ijxk This is called the Lie-Poisson structure on g. 1.2 Classes of Poisson Structures De(cid:27)nition 3. Suppose M is a Poisson manifold. The bracket is called a constant Poisson structure if for every point, there is a neighborhood with local coordinates (x1, . . . , xn) so that {xi, xj} are constants for all i, j. We will call such coordinates canonical coordinates. De(cid:27)nition 4. Suppose M is a Poisson manifold. The bracket is called a log-canonical Poisson 2 structure if for every point, there is a neighborhood with local coordinates (x1, . . . , xn) so that {xi, xj} = ωijxixj for some constants ωij. The coordinates xi are called log-canonical coordi- nates. This is also sometimes called a diagonal Poisson structure. Remark. The reason for the name “log-canonical” is the following. Let x1, . . . , xn be log- canonical coordinates with {xi, xj} = ωij xixj, and consider the functions yi = log(xi). Then we have {yi, yj} = {log(xi), log(xj)} = {xi, xj} = ωij 1 xixj So the functions yi give a canonical system of coordinates. Remark. Let Ω = (ωij) be any skew-symmetric matrix of constants. Then we may de(cid:27)ne a constant Poisson structure on the a(cid:28)ne space Kn by {xi, xj} = ωij. We may also de(cid:27)ne a log-canonical Poisson structure by {xi, xj} = ωijxixj. Log-canonical Poisson structures appear in various areas of mathematics: • Their quantizations give non-commutative algebras that are referred to as quantum a(cid:28)ne space [BG12]. • The decorated Teichmüller space of a surface S (possibly with boundary, and marked points) has a coordinatization in terms of Penner coordinates (or λ-lengths). There is a standard Poisson structure on Teichmüller space (the Weil-Petersson structure) which is log- canonical in the Penner coordinates [Pen12]. • For a surface S and a Lie group G, the character variety RepG(S) := Hom(π(S), G)/G is the space of representations of the fundamental group into G, modulo conjugation. It has a canonical Poisson structure (the Goldman bracket [Gol86]) de(cid:27)ned in terms of the trace functions tr(γ), for γ ∈ π1(S). It is not quite log-canonical, but formally resembles 3 a log-canonical bracket. The case G = PSL2(R) gives the Teichmüller space mentioned above. In this special case, the bracket is in fact log-canonical. • Any cluster algebra possesses a compatible Poisson structure in which every cluster is a system of log-canonical coordinates [GSV10a]. 1.3 Algebraically Reduced Brackets De(cid:27)nition 5. Let X be a Poisson variety, and (x1, . . . , xn) a system of local coordinates such that the structure functions {xi, xj} are all polynomials of the same degree. We call this system of coordinates algebraically reduced if there does not exist any rational change of coordinates (f1, . . . , fn) with the property that the new structure functions {fi, fj} are all polynomials of smaller degree. Example. Consider the a(cid:28)ne plane K2 with coordinates x, y and linear Poisson bracket given by {x, y} = y. This bracket is not algebraically reduced, since we may replace x by z := x , and the bracket becomes y (cid:27) (cid:26) x y {z, y} = , y = {x, y} = 1 1 y In this section, we will prove that log-canonical brackets are algebraically reduced. We (cid:27)rst compute the brackets of Laurent monomials. Lemma 1. Consider the log-canonical bracket on K(x1, . . . , xn) given by {xi, xj} = ωij xixj for some scalars ωij. Let xI denote x n . Then the bracket of two monomials is given by 1 ··· xin i1 {xI , xJ} = ω(I, J) xI+J , 4 where ω(I, J) :=(cid:80) k,(cid:96) ωk(cid:96) ikj(cid:96). Proof. The proof is a simple calculation using the Leibniz rule. Let ek denote the integer vector which is all zeros except a 1 in the kth spot. Then we have k,(cid:96) (cid:88) (cid:88) (cid:88) k,(cid:96) {xI , xJ} = = = ∂xJ ∂x(cid:96) {xk, x(cid:96)} ∂xI ∂xk ikj(cid:96) xI+J−ek−e(cid:96) · ωk(cid:96) xkx(cid:96) ωk(cid:96) ikj(cid:96) xI+J k,(cid:96) The following result was proved by Goodearl and Launois, and is a weaker version of our main theorem. Theorem 1. [GL11] Consider the algebra K(x1, . . . , xn) with log-canonical bracket {xi, xj} = ωij xixj. Then there do not exist rational functions f and g such that {f, g} = 1. Proof. We may embed the (cid:27)eld of rational functions into the (cid:27)eld of iterated Laurent series, and I αI xI, where I is a multi-index (exponent vector). Then express any rational function as f =(cid:80) the bracket of two rational functions f =(cid:80) I αI xI and g =(cid:80) J βJ xJ is given by (cid:88) (cid:88) {f, g} = αI βJ {xI , xJ} = αI βJ ω(I, J)xI+J I,J I,J The constant term of {f, g} will be the sum over those pairs I, J for which I + J = 0. But since ω is skew-symmetric, ω(I,−I) = 0, and so the constant term must be zero. The following result will be useful later. 5 Lemma 2. [MO17] Let P be a Poisson algebra which is also a (cid:27)eld. The following are equivalent. (a) There exist f, g ∈ P with {f, g} = 1. (b) There exist f, g ∈ P with {f, g} = g. (c) There exist f, g ∈ P with {f, g} (cid:54)= 0 and {f,{f, g}} = 0. Proof. (a) ⇒ (b): Suppose that {f, g} = 1. De(cid:27)ne h = f g. Then we have {h, g} = {f g, g} = {f, g}g = g (b) ⇒ (c): Suppose that {f, g} = g. Then {g, f} = −g (cid:54)= 0, but {g,{g, f}} = {g,−g} = 0. (c) ⇒ (a): Suppose that {f, g} (cid:54)= 0, but {f,{f, g}} = 0. De(cid:27)ne h = g{f,g}. Then we have (cid:26) (cid:27) {f, h} = f, g {f, g} {f, g} {f, g} − g {f, g}2{f,{f, g}} = 1 = Finally, we may prove the main theorem. Theorem 2. [MO17] Consider the log-canonical Poisson bracket on K(x1, . . . , xn) given by{xi, xj} = ωij xixj. This Poisson structure is algebraically reduced. Proof. By the theorem of Goodearl and Launois, we already know that there cannot exist any rational functions (f1, . . . , fn) so that {fi, fj} are all constants. It just remains to show that there are no such fi so that {fi, fj} are all linear polynomials in the fi. So suppose that there were such rational functions fi, such that {fi, fj} = ck ijfk (cid:88) k 6 We assume that K is algebraically closed. If not, we may consider the algebra K(x1, . . . , xn) of rational functions with coe(cid:28)cients in the algebraic closure, and we may de(cid:27)ne a Poisson bracket on it using the same skew-symmetric matrix of scalars ωij ∈ K. The expressions for {fi, fj} will be the same as above, and the remainder of the proof will still give the same contradiction. Since the structure functions {fi, fj} are linear, the fi’s span a (cid:27)nite-dimensional Lie algebra. This Lie algebra acts on itself via the adjoint representation, with adf := {f,−}. Consider the linear map adfi for any of the fi. If adfi had any nonzero eigenvalues, then there would be a g such that {fi, g} = λg. By rescaling fi (cid:55)→ λ−1fi, we get {fi, g} = g. By Lemma 2, this means also that there exist some x, y so that {x, y} = 1, which contradicts the theorem of Goodearl and Launois. So it must be that there are no nonzero eigenvalues of adfi . Since K is algebraically closed, this means that adfi is nilpotent. So for any g, there is some k so that adk (g) = 0. fi So again by Lemma 2 this means that there exists x, y such that {x, y} = 1, which is again a contradiction. (g). Then {f, h} (cid:54)= 0, but {f,{f, h}} = adk fi (g) = 0. Let h = adk−2 fi 7 Chapter 2 The Pentagram Map 2.1 Background The pentagram map was (cid:27)rst introduced by Schwartz [Sch92] as a transformation of the moduli space of labeled polygons in the projective plane P2. After labeling the vertices of an n-gon with the numbers 1, . . . , n, the diagonals are drawn which connect vertex i to vertex i + 2 (see Figure 2.1). The intersection points of these diagonals are taken to be the vertices of a new n- gon. Later, Schwartz, along with Ovsienko and Tabachnikov, generalized the map to the space of twisted polygons in P2, and proved that this generalized pentagram map is completely integrable in the Liouville sense [OST10]. In 2011, Max Glick interpreted the pentagram map in terms of Y -mutations of a certain cluster algebra [Gli11]. In the literature, there are other names for Y - mutations or Y -dynamics. Fomin and Zelevinsky have called them coe(cid:28)cient dynamics [FZ02] [FZ07]. Gekhtman, Shapiro, and Vainshtein refer to them as τ-coordinate mutations [GSV10a]. Fock and Goncharov refer to them as X -variable mutations [FG06]. In 2016, Gekhtman, Shapiro, Vainshtein, and Tabachnikov, building on the work of both Glick and Postnikov [Pos06], interpreted a certain set of coordinates on the space of twisted polygons as weights on some directed graph, and the pentagram map in these coordinates as a certain sequence of Postnikov moves applied to this graph [GSTV16]. They used their previous work on the Poisson geometry of the space of edge weights of such graphs [GSV09] [GSV10b] to give a 8 new proof of the integrability of the pentagram map. Recently, Mari Be(cid:29)a and Felipe considered a generalization of the pentagram map, where the twisted polygons were taken to be in the Grassmannian instead of projective space [FMB15], and they demonstrated a Lax representation for this version of the pentagram map, establishing a kind of integrability for this generalized version. The purpose of this dissertation is to inter- pret this Grassmannian pentagram map as a transformation of a set of matrix-valued variables (and more generally as a formal non-commutative rational transformation), and use ideas from non-commutative Poisson geometry – namely the double brackets of Van den Bergh [Ber08] and H0-Poisson structures of Crawley-Boevey[CB11] – to formulate a non-commutative version of in- tegrability, which generalizes the approach of Gekhtman, Shapiro, Vainshtein, and Tabachnikov, using weighted directed graphs. This non-commutative algebraic structure is a generalization of both the original pentagram map and this more recent Grassmann version. Furthermore, the integrability of both examples follows from this non-commutative structure, by projecting to representation spaces of a certain non-commutative algebra. 2.2 Twisted Polygons and the Pentagram Map We refer to the pentagram map as the “classical” pentagram map to distinguish it from the Grass- mann pentagram map (the subject of the later sections). First we give the basic idea, and then extend to so-called “twisted” polygons. The map was originally introduced by Schwartz [Sch92], but here we mainly follow the notations and conventions from [GSTV16]. Consider an n-gon in P2 = RP2, with vertices labeled p1 through pn. Draw the diagonals of the n-gon which connect pi to pi+2 (with indices read cyclically). Label the intersection of the lines pipi+2 and pi+1pi+3 as qi. Then the pentagram map, which we denote by T, sends the (cid:27)rst labeled polygon P to the 9 labeled polygon Q = T (P ) whose vertices are qi. An example for n = 6 is shown below in Figure 2.1. Figure 2.1: The pentagram map for a hexagon p5 q4 p6 q5 q3 p4 q2 p3 p1 q6 q1 p2 More generally, we de(cid:27)ne twisted polygons. De(cid:27)nition 6. A twisted n-gon in P2 is a bi-in(cid:27)nite sequence (pi)i∈Z of points in P2 such that pi+n = M pi for all i, where M is some projective transformation in PGL3(R), referred to as the monodromy of the twisted n-gon. Remark. The usual notion of an n-gon can be thought of as the case where M = Id, and the sequence is periodic. In this case we call the polygon a closed n-gon. The pentagram map, T, can be de(cid:27)ned analogously for twisted n-gons. We always assume twisted n-gons are generic in the sense that no three consecutive points pi, pi+1, pi+2 are collinear. De(cid:27)nition 7. Two twisted n-gons (pi) and (qi) said to be projectively equivalent if there is some G ∈ PGL3(R) so that qi = Gpi for every i. We will denote by Pn the moduli space of twisted n-gons up to projective equivalence. Remark. The pentagram map commutes with projective transformations in the sense that for G ∈ PGL3(R), the polygons T (P ) and T (G · P ) are projectively equivalent. Thus it induces a well-de(cid:27)ned map T : Pn → Pn. 10 There is the following important result about this map: Theorem 3. [OST10] The pentagram map T : Pn → Pn is completely integrable, in the Liouville sense. That is, there exists a T -invariant Poisson structure on Pn and su(cid:28)ciently many independent functions fi ∈ C∞(Pn) for which {fi, fj} = 0. 2.3 Corrugated Polygons and Higher Pentagram Maps In [GSTV16], the authors de(cid:27)ne generalized higher pentagram maps, which we de(cid:27)ne and discuss now. Instead of working in P2, we generalize to Pk−1, with the usual pentagram map being the specialization to k = 3. In the same way as before, we de(cid:27)ne twisted polygons in Pk−1. De(cid:27)nition 8. A twisted n-gon in Pk−1 is a bi-in(cid:27)nite sequence of points (pi)i∈Z in Pk−1 with monodromy M ∈ PGLk so that pi+n = M pi for all i. De(cid:27)ne Pk,n to be the set of all projective equivalence classes of twisted n-gons in Pk−1 with the genericity condition that any consecutive k points do not lie in a proper projective subspace. Keeping with the previous notation, we de(cid:27)ne Pn := P3,n. De(cid:27)nition 9. A corrugated polygon is a twisted n-gon in Pk−1 with the additional property that pi, pi+1, pi+k−1, pi+k span a projective plane for all i. In other words, for any lift vi of pi to Rk, the four vectors vi, vi+1, vi+k−1, vi+k span a 3-dimensional subspace. In particular, when k = 3, any twisted polygon is automatically corrugated. to be the subset of corrugated polygons with the property that for De(cid:27)nition 10. De(cid:27)ne P0 each i, any 3 of the 4 points pi, pi+1, pi+k−1, pi+k are not collinear. In other words, for any lift vi, and any i, any 3 of the 4 vectors vi, vi+1, vi+k−1, vi+k are linearly independent. k,n 11 De(cid:27)ne Li to be the line containing pi and pi+k−1. Then since pi, pi+1, pi+k−1, pk span a projective plane in Pk−1, the lines Li and Li+1 must intersect. We call the intersection point qi. We can then de(cid:27)ne a pentagram map T : P0 k,n → Pk,n which sends (pi) to (qi). Note the codomain is not P0 , since the image my be degenerate. k,n 2.4 Coordinates on the Moduli Space Next we will construct a system of coordinates on Pn, following the presentation in [GSTV16]. The following construction will also give a system of coordinates on P0 for the generalized higher pentagram maps, but we present only the case k = 3, for the sake of simplicity. Given a twisted n-gon (pi), we lift it to a bi-in(cid:27)nite sequence of vectors (vi) in R3. That is, vi projects to pi under the canonical map R3 \ {0} → P2. The genericity assumption guarantees that for each i, the set {vi, vi+1, vi+2} is a basis for R3. Thus we have for each i a linear dependence relation: k,n vi+3 = aivi + bivi+1 + civi+2 (2.1) We call the lift twisted if vi+n = Avi, where A ∈ GL3 is any matrix representing the monodromy. If the lift is twisted, then the sequences ai, bi, and ci are n-periodic, and none of the ai, bi, or ci are zero. The following result is proved in more generality in [GSTV16], but we prove it here for the sake of presentation. Proposition 1. [GSTV16] The lift (vi) can be chosen so that ci = 1 for all i, and the remaining coe(cid:28)cients are n-periodic. Proof. We (cid:27)rst start by choosing a twisted lift (vi), which is always possible. Any other lift (ˆvi) di(cid:29)ers by rescaling. That is, there are non-zero constants λi so that ˆvi = λivi. Then Equation 12 2.1 becomes: (cid:18) ˆvi+3 = ai (cid:19) λi+3 λi (cid:18) ˆvi + bi (cid:19) λi+3 λi+1 (cid:18) ˆvi+1 + ci (cid:19) λi+3 λi+2 ˆvi+2 We want to choose λi so that ci gives the recurrence λi+1 = λi ci−2 recurrence. = 1 for all i. Re-arranging this equation, and re-indexing λi+3 λi+2 . We may therefore set λ0 = 1, and then de(cid:27)ne the rest by this Solving this recurrence gives λi+n = . The factors of c1 ··· cn therefore cancel in the expressions of the new coe(cid:28)cients, and since ai, bi, ci were periodic, so are the new coe(cid:28)cients. λi c1···cn With the previous result in mind, we change notation and let xi := bi and yi := ai, so that Equation 2.1 becomes vi+3 = yivi + xivi+1 + vi+2 (2.2) Lemma 3. [GSTV16] dimP0 k,n = 2n, with the xi, yi being a system of coordinates. Next we will write the pentagram map in these coordinates, and see that it is a rational map. Since we consider the pentagram map acting on labeled polygons, we will abuse notation slightly and write T (pi) = qi for the individual vertices of the polygon. Taking the abuse a step further, if we have lifts (vi) of (pi) and (wi) of (qi) = (T (pi)), we will also write T (vi) = wi. To see how the pentagram map transforms the xi and yi coordinates, we will look at how the pentagram map acts on a lift. First let’s introduce some notation. As before, let Li be the line between pi and pi+2. Recall that the pentagram map is given by qi = T (pi) = Li ∩ Li+1. Let Pi be the plane in R3 which projects onto Li. Then any vector in the intersection Pi ∩ Pi+1 can be taken to be the lift of the image wi = T (vi). In particular, re-arranging Equation 2.2 gives a candidate which can be written in two ways: 13 vi+3 − xivi+1 = yivi + vi+2 ∈ Pi ∩ Pi+1 (2.3) Proposition 2. [GSTV16] The pentagram map T : Pn → Pn is given in the xi, yi coordinates by xi (cid:55)→ xi yi (cid:55)→ yi+1 xi+2 + yi+3 xi + yi+1 xi+2 + yi+3 xi + yi+1 Proof. The lifts of the image polygon’s vertices will also satisfy the linear dependence relation as in Equation 2.1: T (vi+3) = YiT (vi) + XiT (vi+1) + ZiT (vi+2) In the above equation, substitute for T (vi) and T (vi+2) the expressions from Equation 2.3 be- longing to Pi+1, and substitute for T (vi+1) and T (vi+3) the corresponding expressions belong- ing to Pi. Doing so, one obtains the equation vi+5 + yi+3vi+3 = (Xiyi+1 − Yixi) vi+1 + (Xi + Yi − Zixi+2) vi+3 + Zivi+5 Comparing coe(cid:28)cients on both sides, we can solve to get that Zi = 1 Xi = xi Yi = yi+1 xi+2 + yi+3 xi + yi+1 xi+2 + yi+3 xi + yi+1 14 Chapter 3 Networks and Poisson Structures In this section, we review the relevant de(cid:27)nitions and constructions needed to formulate the pentagram map in terms of edge-weighted directed graphs. First, we review the types of weighted directed graphs we will be working with, and an important class of transformations of such graphs. Then we review the Poisson structures introduced in [GSV09][GSV10b] on the space of weights of such graphs. 3.1 Weighted Directed Fat Graphs De(cid:27)nition 11. A fat graph is a graph, together with a prescribed cyclic ordering of the half-edges incident to each vertex. In particular, any graph embedded onto a two-dimensional oriented surface is naturally a fat graph, with the orientation induced from that of the surface. Later on, we will only be considering such graphs which are embedded on a surface, so all graphs will assumed to be fat graphs, even if not explicitly stated. De(cid:27)nition 12. A quiver (or directed graph) is a tuple Q = (Q0, Q1, s, t) where Q0 is the set of vertices of the underlying graph, Q1 the set of edges, and s, t : Q1 → Q0 are the “source” and “target” maps, indicating the direction of the arrows. De(cid:27)nition 13. We will use the term network to mean a weighted directed fat graph. A weighting 15 means an assignment Q1 → R of a real number to each edge. Postnikov considered what he called perfect planar networks [Pos06], which are networks embedded in a disk such that 1. All vertices on the boundary are univalent 2. All internal vertices are trivalent and are neither sources nor sinks 3. All edge weights are positive real numbers Note that the second condition, that internal vertices are trivalent, and are not sources or sinks, implies that they are one of two types: either a vertex has one incoming and two outgoing edges, or it has one outgoing and two incoming edges. When we draw networks, we will picture the former as white vertices, and the latter as black. This is pictured in Figure 3.1. Figure 3.1: Types of vertices in a perfect network De(cid:27)nition 14. [Pos06] For an acyclic perfect network Q, let a1, . . . , ak denote the sources on the boundary, and let b1, . . . , b(cid:96) be the sinks on the boundary. The boundary measurement bij is de(cid:27)ned to be the sum of the weights of all paths from ai to bj, where the weight of a path is the product of the weights of its edges. We arrange the boundary measurements into a k × (cid:96) matrix B(Q) = BQ = (bij), called the boundary measurement matrix of Q. Later on, we will be interested only in a particular acyclic network, so we will not discuss the more general de(cid:27)nition for when Q has directed cycles. 16 Postnikov described several local transformations of networks which leave the boundary mea- surements invariant. The moves are pictured in Figure 3.2, and are labeled with the correspond- ing edge weights. Figure 3.2: Local Postnikov moves (∆ := b + adc) ( I ) ( II ) ( III ) d b e4 c e3 e1 a e2 x a b c y x a b c y e1 d b ∆ d c ∆ e4 d a ∆ e2 ∆ e3 x b b yb xb b y b c c a a We refer to the type I move as the “square move”, the type II move as “white-swapping”, and the type III move as “black-swapping”. It is an easy exercise to check that these three moves do not change the boundary measurements from a source to a sink in these pictures. There is another type of local move called a gauge transformation, which changes the edge weights but not the underlying graph. Let λ be any non-zero number. Then at any vertex we may multiply all incoming weights by λ and all outgoing weights by λ−1. This obviously does not change the boundary measurements. The group generated by all gauge transformations is 17 called the gauge group. From now on, we will only consider networks where the sources and sinks are not interlaced, so that all sources can be pictured on the left side, and all sinks on the right. The reason for restricting to these networks is that they have the following nice property. One may consider “concatenating” two such networks in a disk, by glueing segments of their boundaries in a way that each sink on one is identi(cid:27)ed with a source of the other. This is pictured in Figure 3.3. The resulting edge after identi(cid:27)cation is weighted by the product of the two edges. If the sources and sinks are not interlaced, as mentioned above, then the boundary measurement matrix of the resulting network after concatenation is the product of the two boundary measurement matrices. We may also consider weighted directed networks embedded on an annulus [GSV10b]. Later, when we talk about the pentagram map, it will be these annular networks that we will consider. For simplicity, we will again only consider acyclic networks. Similar to the disk case, consider perfect networks, meaning any internal vertices are trivalent (and not sources or sinks), and that all boundary vertices are univalent. We also assume all vertices on the inner boundary component are sources, and all vertices on the outer boundary component are sinks. Then in a similar way as before, we de(cid:27)ne a boundary measurement matrix, whose entries are the sums of path weights from a given source to a given sink. Since the Postnikov moves are local, and do not take into account the global topology of the surface on which the network is embedded, they may also be applied in this case, and of course they still do not change the boundary measurements. As in the disk case, we may consider concatenation of two annular networks, by identifying the outer boundary circle of one with the inner boundary circle of the other, in a way that each sink is identi(cid:27)ed with a source. If we assume, as mentioned above, that all sources are on the inner boundary circle and all sinks on the outer, then the resulting boundary measurement matrix is the product of the two boundary measurement matrices. 18 If there are the same number of sources and sinks, then it will sometimes be convenient to glue the inner and outer boundary circles together so that pairs of sources and sinks are identi(cid:27)ed, to obtain a network on the torus. Lastly, we de(cid:27)ne the modi(cid:27)ed edge weights, which are elements of the Laurent polynomial ring R[λ±] in the indeterminate λ, de(cid:27)ned as follows. Choose an oriented curve ρ, called the cut, which connects the inner and outer boundary components of the annulus. Suppose an oriented edge α ∈ Q1 has weight xα ∈ R. It may happen that α intersects the cut. If i is an intersection point, de(cid:27)ne εi = 1 if the tangent vectors to α and ρ at i form an oriented basis of the tangent space, and de(cid:27)ne εi = −1 if they have the opposite orientation. Then the modi(cid:27)ed edge weight of α is de(cid:27)ned as xαλεi If we use the modi(cid:27)ed edge weights, then the boundary measurements become Laurent polyno- mials in λ. From now on, we always assume the boundary measurements are in terms of modi(cid:27)ed edge weights. We will use the notation BQ(λ) to emphasize that the entries are functions of λ. Remark. As mentioned above, if there are the same number of sources and sinks, we may glue the two boundary circles together to obtain a network on the torus. The two boundary circles become a single loop on the torus, which we call the rim. If we choose a new rim, and cut the torus along this new rim, we get a new (possibly di(cid:29)erent) network on an annulus. The boundary measurement matrices di(cid:29)er by a re-factorization, which we will now explain. If we draw the new rim on the old annulus, we can realize the annulus as the concatenation of two annular networks Q(cid:48) and Q(cid:48)(cid:48), one on each side of the new rim. Let A and B be the boundary measurement matrices of Q(cid:48) and Q(cid:48)(cid:48) respectively. Then the boundary measurement matrix is AB. Glueing into a torus and then cutting along the new rim amounts to concatenating the pictures of Q(cid:48) and Q(cid:48)(cid:48) in the 19 opposite order. The new boundary measurement matrix is thus BA. It was observed by Izosimov [Izo18] that many di(cid:29)erent generalizations of the pentagram map may be described in terms of these types of matrix re-factorizations in Poisson Lie groups. 3.2 Poisson Structures on the Space of Edge Weights Given a perfect network in an annulus, as in the last section, we may forget the particular choice of edge weights and consider the underlying quiver Q = (Q0, Q1). The space of edge weights of the quiver Q, denoted EQ := (R∗)Q1, is the space of all possible choices of non-zero weights. Gekhtman, Shapiro, and Vainshtein de(cid:27)ned a family of Poisson structures on EQ [GSV09], which we will review now. The Poisson structures are de(cid:27)ned locally at each vertex, and then the local Poisson brackets are shown to be “compatible” with concatenation (this will be made more precise later), and so they can be combined to give a global Poisson bracket. This is outlined in [GSV09] and [GSV10b], but we present the construction here in detail, since we will mimic it very closely later on when we de(cid:27)ne a non-commutative analogue. Recall that for networks on an annulus, we require internal vertices are trivalent, and that they are neither sources nor sinks, and so they are either white or black, as mentioned before: z x y c a b Let E◦ be the space (R∗)3 with coordinates x, y, z. We will think of the variables x, y, z as representing the edge weights at a white vertex picture above. Similarly, let E• = (R∗)3 with coordinates a, b, c corresponding to a black vertex. Of course they are di(cid:29)eomorphic, but we will 20 de(cid:27)ne di(cid:29)erent Poisson brackets on them. Choose any w1, w2, w3 ∈ R, and de(cid:27)ne a log-canonical Poisson bracket {−,−}◦ on E◦ by {x, y}◦ = w1xy, {x, z}◦ = w2xz, {y, z}◦ = w3yz Similarly, for scalars k1, k2, k3 ∈ R, de(cid:27)ne {−,−}• on E• by {a, b}• = k1ab, {a, c}• = k2ac, {b, c}• = k3bc Next we will consider the operation of glueing/concatenating these local pictures together, as described before. This means identifying part of the boundary of one picture with part of another, in such a way that a sink is identi(cid:27)ed with a source. We then erase the common glued boundary, and remove the common identi(cid:27)ed vertex, identifying the two incident edges (which have a consistent orientation by construction). The weight of the new edge is de(cid:27)ned to be the product of the weights of the edges being glued. Figure 3.3 illustrates glueing a black and white local picture: Figure 3.3: Concatenation of local pictures c z a b x y (cid:32) c b z y ax The de(cid:27)nitions of {−,−}◦ and {−,−}• are “compatible” with this glueing procedure in a way that we now make precise. Recall that EQ = (R∗)Q1 is the space of edge weights. We can associate to it the algebra EQ := R(Q1), which is the (cid:27)eld of rational functions with indetermi- (cid:96) Q1 to be the nates corresponding to the arrows of the quiver. We also de(cid:27)ne HQ := (R∗)Q1 21 space of half-edge weights. This is an assignment of a non-zero number to each half-edge. Each half-edge is associated to the vertex which it is incident to. For an arrow α ∈ Q1, we call the corresponding half-edges αs and αt, for the source and target ends of the arrow. The algebra of functions HQ corresponding to HQ is then the (cid:27)eld of rational functions in twice as many variables, corresponding to αs and αt for α ∈ Q1. Recall that Q0 denotes the set of vertices of the quiver Q. We can partition this into Q0 = V◦ ∪ V• ∪ V∂ = Vi ∪ V∂, where V◦ is the set of white vertices, V• the black vertices, and V∂ the boundary vertices, and Vi = V◦ ∪ V• is the set of internal vertices. Note that since all internal vertices are trivalent, and all boundary vertices are univalent, the set of half-edges is in bijection (cid:96) Vi. Recall we de(cid:27)ned the spaces E◦ ∼= E• ∼= (R∗)3. We now additionally (cid:96) Vi (cid:96) Vi ∼= R∗ with trivial Poisson bracket, corresponding to boundary vertices. Then the space with V∂ de(cid:27)ne E∂ of half-edges can be thought of as ∼= E |V∂| ∂ × E |V◦| ◦ × E |V•| • HQ Under this identi(cid:27)cation, we realize the algebra HQ as the tensor product ∼= E ⊗|V∂| ∂ HQ ⊗|V◦| ⊗ E ◦ ⊗|V•| ⊗ E • From the Poisson brackets de(cid:27)ned above on the algebras E◦, E•, and E∂, we get a natural induced Poisson bracket on HQ, where {α, β} = 0 if α and β are half-edges which are not incident to a common vertex, and the boundary half-edges are casimirs. (cid:55)→ We de(cid:27)ne a “glueing” map g : HQ → EQ, represented by Figure 3.3, given by (αs, αt)α∈Q1 . The earlier claim that the local brackets on E◦ and E• are compatible with glueing (αsαt)α∈Q1 22 is made precise by the following. Proposition 3. There is a unique Poisson bracket on EQ such that the glueing map g : HQ → EQ is a Poisson morphism. Proof. The statement that g is a Poisson map is equivalent to the pull-back map g∗ : EQ → HQ being a homomorphism of Poisson algebras. Note that for α ∈ Q1, the map g∗ is given by g∗(α) = αsαt. To be a Poisson algebra homomorphism would mean that for any α, β ∈ Q1, {g∗(α), g∗(β)}HQ = g∗(cid:16){α, β}EQ (cid:17) We claim that {α, β}EQ is determined by this property. If α and β do not share a common vertex, then obviously {α, β} = 0. There are still many cases to consider, depending on which end (source or target) of each α and β meet at the common vertex, and where in the cyclic ordering those half-edges are at that vertex. For example, let’s consider the picture in Figure 3.3, and denote by f the new edge with weight ax. Let’s try to de(cid:27)ne {f, y}EQ . We only see one end of y in the (cid:27)gure, so let’s say g∗(y) = ysyt, where ys is the end we see at the white vertex. Then the condition that g∗ be Poisson means we must have g∗(cid:16){f, y}EQ (cid:17) = {g∗(f ), g∗(y)}HQ = {ax, ysyt}HQ = a{x, ys}HQ yt = w1 axysyt = w1g∗(f )g∗(y) = g∗ (w1 f y) 23 Since g∗ is injective, this uniquely de(cid:27)nes {f, y}EQ . The calculations for all other cases are similarly simple. We see that for α, β ∈ Q1 with a common vertex, the bracket {α, β}EQ given by the same expression as the bracket of the corresponding half-edges in HQ. is We will also want to consider the doubled quiver. De(cid:27)nition 15. The doubled quiver, Q, is de(cid:27)ned to have the same vertex set as Q, whereas the edge set of Q is the disjoint union of two copies of Q1. For each arrow α ∈ Q1, there are two arrows α and α∗ in Q1. The arrow α∗ is called the “opposite” arrow of α, with s(α∗) = t(α) and t(α∗) = s(α). We associate α∗ with the function α−1 on EQ. Then a path in Q may be represented as a Laurent monomial, which is the product of the edge weights along the path (allowing inverses). Since Poisson brackets extend uniquely to localizations, we may extend the Poisson bracket de- (cid:27)ned above to all Laurent polynomials (and indeed any rational functions) on EQ. We will now be interested in describing the Poisson bracket of two paths, thought of as ra- tional functions on EQ. To do so, it will be convenient to introduce the constants A• = k1 − k2 − k3 A◦ = w1 − w2 − w3 Consider two paths f and g. Whenever the paths meet (share at least one edge in common), then there is a corresponding maximal subpath which f and g share. If the paths go in the same direction on the common subpath, we will say they are parallel on that common subpath. If f and g are parallel on a proper subpath, then there are two possibilities, which are depicted in Figure 3.4. In the (cid:27)gure, f is the blue path, g is the red path. In the (cid:27)rst situation (the left image in the 24 (cid:27)gure), the paths are said to “touch”, and in the second situation, they are said to “cross”. The other possibility is that one of the paths (either f or g) is a subpath of the other. In this case, if f is a subpath of g, then we may write g = f g(cid:48), where g(cid:48) is the rest of g. We extend the notions of “touching” and “crossing” to the situation where g = f g(cid:48) by saying f and g “touch” on the subpath f if g(cid:48) and f touch, and similarly for “crossing”. Figure 3.4: Two parallel paths sharing a common subpath ··· “touching” ··· “crossing” We will denote by f ∩ g the set of all maximal common subpaths of f and g. We will de(cid:27)ne a function ε(f, g) on the set f ∩ g, and for x ∈ f ∩ g, we denote the value by εx(f, g). It depends on whether f and g touch or cross, and on the colors of the vertices at the endpoints of the common subpath. The values are given in Table 3.1. Table 3.1: De(cid:27)nition of ε(f, g) for parallel paths type touch touch touch touch cross cross cross cross left endpt • ◦ ◦ • • ◦ ◦ • right endpt • ◦ • ◦ • ◦ • ◦ εx(f, g) 0 0 −(A◦ + A•) A◦ + A• 2A• −2A◦ A• − A◦ A• − A◦ We de(cid:27)ne ε(f, g) to be skew-symmetric, so that ε(g, f ) = −ε(f, g) if the roles are switched. Also, there are other cases of intersecting paths which we have not considered. If we allow paths in Q, then two paths can meet with opposing orientations on a common subpath. This can be obtained from the local pictures above by changing one of the paths to its inverse. Since Poisson 25 brackets extend uniquely to localizations, this will not be an issue. We then have the following formula: Proposition 4. Suppose f and g are two paths. Then (cid:88) ∗∈f∩g {f, g} = ε∗(f, g) f g Proof. Since f and g are monomials in the coordinate variables, it follows from Lemma 1 that {f, g} = c f g for some constant c. We must prove that c = (cid:80) ε∗(f, g). Each time f and g share a proper subpath on which they are parallel, it will look like one of the two pictures in Figure 3.4. We can then write the paths as f = f0axwyb and g = g0cxwyd, where xwy is the common subpath, a,x, and c are the edges incident to the “left” vertex (where the paths come together), and y,b, and d are the edges incident to the “right” vertex (where the paths diverge at the end of the common subpath). Then after expanding {f, g} by the Leibniz rule, we get contributions from {a, c}, {a, x}, and {x, c} at the left vertex, and {y, d}, {b, y}, and {b, d} from the right vertex. Obviously the contributions from the common subpath cancel, since if p and q are two consecutive edges in the common subpath, we will get a contribution from both {p, q} and {q, p}. There are of course other terms in the expansion coming from f0 and g0, but they will correspond to other common subpaths in the same way. It is therefore enough to consider the six terms mentioned above (3 each for the beginning and end vertices) for each common subpath. In general, we must consider 36 di(cid:29)erent cases, since each endpoint could be either white or black, and there are three orientations for the arrows incident to each of the endpoint vertices. A simple calculation shows that the orientations at the vertices are in fact irrelevant. Therefore we only have 4 cases, depending on the colors of the vertices. As mentioned above, by the Leibniz rule, we can add the contributions from the endpoint vertices separately. 26 First consider the left endpoint. For all three choices of orientations, a black left endpoint will give a contribution of A• to the coe(cid:28)cient, and a white left endpoint will give a contribution of −A◦. For the right endpoint, a black vertex gives −A•, and a white gives A◦. Combining all the possibilities together gives the list of values in the table for εx(f, g). We assumed so far that the common subpaths were parallel. If the paths are oriented in opposite directions on a common subpath, then it corresponds to reversing one of the paths in Figure 3.4. Then we can extend the table for εx(f, g) by the rule {f, g−1} = −g−2{f, g}. We also assumed so far that the subpaths were proper. It remains to consider when one of the paths is a subpath of the other. For instance, let’s say f is contained in g. Then we can write g as g = f g(cid:48). Then by the Leibniz rule, {f, g} = {f, f g(cid:48)} = f{f, g(cid:48)}. It must be (since all vertices are trivalent) that the beginning of g(cid:48) is a common subpath with f, and so this is the term corresponding to the common subpath of f and g which is all of f. The rest of the common subpaths of f and g(cid:48) are just usual proper common subpaths of f and g, as discussed above. Remark. Note that this result depends only on the parameters A◦ and A•, and not on the actual values of w1, w2, w3, k1, k2, k3. In particular, we could choose w1 = w2 = k1 = k2 = 0, and w3 = −A◦, k3 = −A•, and the formula from the proposition would be the same. From now on, we will assume this is the case (that all but w3 and k3 are zero). 27 Chapter 4 Interpreting the Pentagram Map in Terms of Networks We now use the constructions from the previous section to realize the pentagram map as a se- quence of Postnikov moves and gauge transformations of a particular network in an annulus. The properties of the boundary measurements will give us invariants of the pentagram map, which also turn out to be involutive with respect to the Poisson structures described in the previous section. This section summarizes the main points of [GSTV16]. 4.1 The Quiver and Poisson Bracket We now look at a very speci(cid:27)c example of the quivers and Poisson structures discussed previously. The quiver Qk,n (or just Qn if k = 3) is embedded in an annulus (or cylinder), and the case k = 3, n = 5 is shown below in Figure 4.1. In general, there are k sources and k sinks, the number of square faces is n, and they are connected as in the (cid:27)gure. More speci(cid:27)cally, the bottom right of the ith square face connects to the top left of the (i + 1)st, and the top right of the ith square face is connected to the bottom left of the (i+2)nd, with the indices understood cyclically. This is how the labels of the sinks on the right side are chosen. The top and bottom edges of the boundary rectangle are identi(cid:27)ed, giving a cylinder. We take the cut to be the top and bottom edges which we identify. The sources and sinks are labeled on the left and right edges, which become the 28 inner and outer boundary circles, respectively. As was mentioned in the previous sections, we will sometimes consider the network on a torus by also identifying the left and right edges (up to a twist, indicated by the labels). Figure 4.1: The network Q3,5 1 2 3 2 3 1 As was mentioned in the previous section, the Poisson bracket depends only on the parame- ters A◦ and A•. So we may assume that all wi and ki are zero except w3 = −A◦ and k3 = −A•. . Note that when two paths meet From now on we will consider the speci(cid:27)c choice w3 = k3 = 1 in this network, they must come together at a black vertex, and separate at a white vertex. This means for two paths f and g, that εx(f, g) = ±1 when f and g touch, and εx(f, g) = 0 when f and g cross. 2 4.2 The x, y Coordinates Next, we will apply gauge transformations to this network and consider some other functions on EQ. We start by de(cid:27)ning our notation for the edge weights. We will consider the toric network obtained by glueing the boundary components together. So the nth square connects to the 1st square. We label the edge weights around each square face as follows: 29 εi−1 αi εi−2 δi εi γi βi εi Because of our choice of coe(cid:28)cients A◦ = A• = − 1 2 , the brackets between these coordinates are given by {βi, αi} = {αi, εi−1} = 1 2 1 2 βiαi εi−1αi {βi, γi} = {γi, εi} = 1 2 1 2 γiβi γiεi After applying gauge transformations at the corners of each square face, we can obtain the following edge weights: εi−1δiεi αiε−1 i−1 ε−1 i γi βi The e(cid:29)ect of these gauge transformations is that all the edges other than those bounding the square faces have been set to 1. These new weights, which are monomials in the original weights, we will call by ai, bi, ci, di: ai = αiε−1 i−1 ci = ε−1 i γi bi = βi di = εi−1δiεi 30 It is easily checked that the Poisson brackets of these monomials are given by {bi, ai} = {ai, di} = 1 2 1 2 biai aidi {bi, ci} = {ci, di} = 1 2 1 2 bici cidi We will now apply further gauge transformations, in order to set as many edge weights as possible equal to 1. It is possible to set most of the weights equal to 1, except the bottom and left edges of each square face, and a few other edges. The result (again for n = 5, k = 3) is pictured in Figure 4.2. Figure 4.2: The quiver Q3,5 after gauge transformations x1 y1 1 2 3 x2 y2 x3 y3 x4 y4 z z z 2 3 1 x5 y5 The weights in the new quiver after gauge transformations are given as follows. ai ci−1di−1ci−2 bi cidici−1di−1ci−2 xi = yi = z = n(cid:89) dkck k=1 Thinking of the quiver as being on a torus, all of these monomials represent loops. They are depicted in Figure 4.3. The blue loop is x4, the red loop is y4, and the green loop is z. 31 Figure 4.3: The weights xi, yi, z represented as cycles 1 2 3 2 3 1 It is an easy calculation to see that z is a Casimir of the Poisson bracket, and so the bracket descends to the level surface in EQ de(cid:27)ned by z = 1. We will now consider just edge weights lying on this hypersurface. In this case, we have that all edge weights except those labeled by xi and yi are equal to 1. Therefore, this level surface has coordinates given by xi and yi, and so it is 2n-dimensional. We will associate these coordinates with the xi, yi coordinates on Pn introduced in section 1. Their brackets are given by {xi+1, xi} = xi+1xi {yi+1, yi} = yi+1yi {yi+2, yi} = yi+2yi {yi, xi} = yixi {yi, xi−1} = yixi−1 {xi, yi−1} = xiyi−1 {xi, yi−2} = xiyi−2 4.3 The p, q Coordinates It will be convenient to also consider a di(cid:29)erent set of functions on EQ. We return to the a, b, c, d variables before the gauge transformations which gave the x, y coordinates. We de(cid:27)ne the face weights pi and qi as the counter-clockwise paths around the faces of the quiver, taking inverses 32 when the orientation disagrees: pi = bi aicidi qi = ci−2di−1ai+1 bi These are related to the x, y coordinates by pi = yi xi qi = xi+1 yi De(cid:27)ne the quiver Q∨, dual to Q, to have vertices corresponding to the faces of Q, and arrows corresponding to arrows of Q connecting vertices of di(cid:29)erent colors, directed so that white ver- tices are on the left and black vertices on the right. The vertices of Q∨ are labeled by pi and qi, with arrows pi → qi−1, pi → qi+2, qi → pi, and qi+1 → pi. This is pictured in Figure 4.4, with the dual quiver drawn in blue. Figure 4.4: The dual quiver Q∨ qi−1 qi+1 pi qi qi+2 The brackets of the p’s and q’s is also log-canonical, with the skew-symmetric coe(cid:28)cient matrix given by the adjacency matrix of the dual quiver Q∨. That is, {qi, pi} = qipi {pi, qi−1} = piqi−1 {qi+1, pi} = qi+1pi {pi, qi+2} = piqi+2 33 4.4 The Postnikov Moves and the Invariants Now we will describe a sequence of Postnikov moves which will transform this quiver (considered as being on the torus) into an isomorphic quiver. The new edge weights obtained after this sequence will be the expressions for the pentagram map on Pn in the x, y coordinates given in chapter 1. We start with the quiver as in Figure 4.2, where all weights are 1 except the xi and yi weights on the bottom and sides of the square faces. We then apply the following moves, in order: 1. Perform the “square move” at each of the n square faces 2. Perform the “white-swap” at each white-white edge 3. Perform the “black-swap” at each black-black edge After this sequence of moves, the underlying directed graph is isomorphic to the one we started with. However, the edge weights will not be of the same form. It remains to perform gauge transformations (as we did in the previous section) so that again all weights are 1 except the bottom and left of each square face. After these gauge transformations, and after choosing a particular choice of graph isomorphism, the x, y weights transform as xi (cid:55)→ xi yi (cid:55)→ yi+1 xi+2 + yi+2 xi + yi xi+3 + yi+3 xi+1 + yi+1 This is almost (but not quite) the same as the expression for the pentagram map on Pn given in section 1. But after making the change of variables yi (cid:55)→ yi−1, the formulas agree. So it only di(cid:29)ers by a shift of indices in the y-variables. 34 As discussed before, the Postnikov moves and gauge transformations performed above do not change the boundary measurements. However, since the quiver was considered on a torus, we may have had some of the vertices or edges “wrap around” from the right side to the left side when doing the “white-swap” and “black-swap” moves. This corresponds to cutting a piece o(cid:29) the right side of the picture, and glueing it back onto the left side. As discussed earlier, this changes the boundary measurement matrix by a re-factorization B1B2 (cid:55)→ B2B1. The boundary measure- ment matrix BQ(λ) is therefore changed only up to conjugation, since B2B1 = B2(B1B2)B−1 . Thus the components of the characteristic polynomial χ(t) = det(Id+tB(λ)) are unchanged by the sequence of moves described above. Recall that we consider the elements of the boundary 2 measurement matrix to be the modi(cid:27)ed edge weights, which are Laurent polynomials in the variable λ. So χ is a function of both λ and t, which is polynomial in t, and Laurent in λ. We denote the coe(cid:28)cients (which are functions of the edge weights of the quiver) by Iij: (cid:88) χ(λ, t) = Iijλitj i,j The discussion above implies that Iij are invariants of the pentagram map. Furthermore, we have: Theorem 4. [GSTV16] Let Iij be as de(cid:27)ned above. Then (a) (b) Iij are invariant under the pentagram map {Iij, Ik(cid:96)} = 0 for all i, j, k, (cid:96) (c) The pentagram map is completely integrable We will also consider the dynamics in the p, q coordinates described before. A simple calcula- tion using the relations between p, q and x, y shows that the pentagram map transforms the p, q coordinates by 35 qi (cid:55)→ 1 pi+1 pi (cid:55)→ qi+2 (1 + pi)(1 + pi+3) (1 + p−1 i+1)(1 + p−1 i+2) In these coordinates we can interpret the Postnikov moves as cluster transformations. To see this, consider the dual quiver Q∨ described before. Interpret the p, q variables as the initial cluster of a cluster algebra whose exchange matrix B = (bij) is de(cid:27)ned by the dual quiver Q∨. We use the mutation formula  µk(τi) = 1 τi (cid:18) τi 1 + τ sign(bki) k (cid:19)bki if i = k if bki (cid:54)= 0 These are referred to as cluster X -variables (as opposed to cluster A-variables) by Fock and Goncharov [FG06]. They are also called “τ-coordinates” by Gekhtman, Shapiro, and Vainshtein [GSV10a]. This type of cluster algebra is called a cluster Poisson algebra, since it inherits a natural Poisson bracket which is “compatible” with mutation (see [GSV10a]), given by {τi, τj} = bijτiτj Using the dual quiver Q∨ for the exchange matrix B, this cluster Poisson bracket in the p, q variables coincides with the bracket on EQ. Additionally, the square moves in the sequence giving the pentagram map coincide with mutations at the p-vertices. The sequence of mutations, µ, 36 which mutates at each pi once, gives µ(pi) = 1 pi µ(qi) = qi (1 + pi+1)(1 + pi−2) )(1 + p−1 (1 + p−1 i−1) i The resulting quiver is the opposite of Q (all arrows are reversed). Additionally performing the white-swap and black-swap Postnikov moves reversed the arrows of the dual, giving a graph isomorphic to the original. This amounts to exchanging the µp and µq face weights. That is, the new square face weights p∗ and the new octagonal face weights q∗ are i i p∗ i = µ(qi) = qi (1 + pi+1)(1 + pi−2) (1 + p−1 )(1 + p−1 i−1) i q∗ i = µ(pi) = 1 pi The pentagram map coincides with this formula up to the permutation of the variables which shifts the indices by p∗ , In other words, we have and q∗ i (cid:55)→ q∗ i+1 i (cid:55)→ p∗ i+2 T (pi) = µ(qi+2) T (qi) = µ(pi+1) It is a simple calculation to verify that the brackets {T (pi), T (qj)} have exactly the same form as {pi, qj}. 37 Chapter 5 The Grassmann Pentagram Map 5.1 Twisted Grassmann Polygons and the Pentagram Map Mari Be(cid:29)a and Felipe studied a generalization of the pentagram map [FMB15] to the Grass- mann manifold. The exposition and notation in this section is largely borrowed from that pa- per, with minor variations. The projective plane P2 coincides with the Grassmannian Gr(1, 3) of 1-dimensional subspaces of R3. A natural generalization would be to consider Gr(N, 3N ), in which the previous case is just when N = 1. In actuality, Mari Be(cid:29)a and Felipe considered more generally Gr(N, kN ), which generalizes Pk−1, but we will focus here on the case k = 3. Consider the set Mat3N×N of 3N-by-N real matrices. There are two natural multiplication actions: on the left by GL3N, and on the right by GLN. If we call MN ⊂ Mat3N×N the subset of rank-N matrices, then we will identify Gr(N, 3N ) with the orbit space MN /GLN by this right action, since two matrices will be equivalent if they have the same column-span. Then Gr(N, 3N ) carries a natural left action by GL3N, induced by the action on Mat3N×N. De(cid:27)nition 16. A twisted Grassmann n-gon (or just “polygon” if n is understood) is a bi-in(cid:27)nite sequence (pi)i∈Z of points in Gr(N, 3N ), with the property that pi+n = M pi for some projective transformation M ∈ PGL3N called the monodromy. As in the classical case, we will only consider twisted polygons satisfying some nondegen- eracy condition. More speci(cid:27)cally, choose a lift (Vi) of (pi) to MN. De(cid:27)ne the block column 38 matrices Vi = (ViVi+1Vi+2). We require for any i that Vi is nonsingular, or equivalently, that the combined columns of Vi, Vi+1, Vi+2 form a basis of R3N. If this nondegeneracy condition is satis(cid:27)ed, then we obtain something analogous to the linear dependence relations given in Equa- tion 2.1. For each i, there are matrices Ai, Bi, Ci ∈ GLN so that Vi+3 = ViAi + Vi+1Bi + Vi+2Ci (5.1) For the remainder of the paper, we will use the abbreviated phrase twisted polygon to mean a twisted Grassmann polygon. We will use the notation GPn,N to denote the moduli space of twisted Grassmann n-gons in Gr(N, 3N ), up to the action of PGL3N. We will use the adjective “classical” when we wish to distinguish the P2 case (N = 1). Next we will describe the pentagram map on Grassmann polygons. Let (pi) be a twisted polygon in Gr(N, 3N ). Since pi and pi+2 are N-dimensional subspaces with trivial intersection, they span a 2N-dimensional (codimension N) subspace, which we call Li. Then the intersection Li ∩ Li+1 is a codimension 2N (dimension N) subspace, which is again an element of Gr(N, 3N ). We de(cid:27)ne the pentagram map to be the map that sends (pi) to (qi), where qi = Li ∩ Li+1. We will also refer to the pentagram map as T : GPn,N → GPn,N. Abusing notation as in the classical case, we also write the map as if it were de(cid:27)ned on individual vertices T (pi) = qi, and also write the map as if it is de(cid:27)ned on lifts T (Vi) = Wi if (Vi) and (Wi) are lifts of (pi) and (qi). 39 5.2 Corrugated Grassmann Polygons and Higher Pentagram Maps Analogous to the classical case, we can also de(cid:27)ne generalized higher pentagram maps in the Grassmann case. This is a generalization from Gr(N, 3N ) to Gr(N, kN ). A twisted Grassmann polygon in Gr(N, kN ) is a sequence (pi) with monodromy M ∈ PGLkN so that pi+n = M pi for all i. We denote the set of all equivalence classes of twisted polygons (up to the action of PGLkN) by GPk,n,N. In keeping with the notation of the previous section, we write just GPn,N for k = 3. Given a polygon P = (pi), we de(cid:27)ne the subspaces Li := pi + pi+k−1. We say that a twisted Grassmann polygon is corrugated if Li + Li+1 is a 3N-dimensional subspace of RkN for all i. Let GP0 denote the set of classes of corrugated polygons with the following additional properties, all of which hold generically: k,n,N • Any 3 of the 4 subspaces pi, pi+1, pi+k−1, pi+k span a 3N-dimensional subspace. • Li ∩ Li+1 ∩ Li+2 = 0 • dim(Li ∩ Li+k−1) = N We de(cid:27)ne the generalized higher pentagram map T : GP0 “corrugated” property guarantees that k,n,N → GPk,n,N as follows. The dim(Li ∩ Li+1) = dim Li + dim Li+1 − dim(Li + Li+1) = 2N + 2N − 3N = N So qi := Li ∩ Li+1 is again an element of Gr(N, kN ), and we de(cid:27)ne the pentagram map to be (pi) (cid:55)→ (qi). 40 Proposition 5. If P = (pi) ∈ GP0 k,n,N , then its image under T is corrugated. Proof. We need to show that for the image T (P ) = Q = (qi), the span qi + qi+1 + qi+k−1 + qi+k is 3N-dimensional. De(cid:27)ne subspaces Ki := qi + qi+1. Then we are trying to show that dim(Ki + Ki+k−1) = 3N. Note that by de(cid:27)nition, qi and qi+1 are both subspaces of Li+1, and so Ki ⊆ Li+1. The assumption that Li ∩ Li+1 ∩ Li+2 = 0 ensures that dim Ki = 2N, and so in fact Ki = Li+1. Then by shifting indices we also get that Ki+k−1 = Li+k. It is always true that Li+1 ∩ Li+k contains pi+k, and so its dimension is at least N. The third assumption, however, guarantees that the intersection is exactly pi+k. We then have that dim(Ki + Ki+k−1) = dim Ki + dim Ki+k−1 − dim(Ki ∩ Ki+k−1) = 3N Remark. It is worth pointing out that the image of the map is not necessarily in the set GP0 k,n,N The above proof shows that the image is corrugated, but it does not necessarily satisfy the gener- . icity assumptions. The set on which the map may be iterated is thus the complement of countably many subsets of codimension 1 (and hence non-empty). 5.3 Description of the Moduli Space by Networks For simplicity of presentation, we will for the remainder of the dissertation restrict to considering the case k = 3, where all polygons are automatically corrugated. In this section, we model the moduli space GPn,N using the networks Qn from Chapter 4. We will use variations of this network on three surfaces: the cylinder S1 × [0, 1] (with two boundary components), the in(cid:27)nite cylinder S1 × (0, 1), and the torus S1 × S1. 41 First, we establish some notation. As before, we let MN be the space of all 3N-by-N matrices of full rank. It is an open subset of Mat3N×N. In particular, for a twisted polygon P ∈ GPn,N, a lift of P will be a point in MZ the subset of lifts of twisted n-gons, and we let T Ln,N denote the set of lifts which are “twisted” (the lifts satisfy Vi+n = M Vi). Since a twisted lift is determined by the (cid:27)rst n of the Vi and the monodromy, we may view T Ln,N as an open subset of Mn N × GL3N. The dimension is thus 3N 2(n + 3). . We denote by Ln,N ⊂ MZ N N Recall the networks Qn de(cid:27)ned in Chapter 4, which are embedded on the cylinder. We will also consider the in(cid:27)nite networks (cid:101)Qn, embedded on the in(cid:27)nite cylinder, which are just sinks. Finally, we also consider the networks (cid:98)Qn, on the torus, which are obtained by glueing in(cid:27)nitely many copies of Qn concatenated together, according to the labels of the sources and the boundary components of the cylinder to each other according to the labels of the sources and sinks in Qn. . ∼= GL Z N Let En,N be the space of all possible choices of GLN weights on the edges which are the ∼= left, bottom, and right of each square face of the network Qn. Clearly there is a bijection En,N . Similarly, let(cid:101)En,N be space of GLN-weights on the same edges, but on the in(cid:27)nite network (cid:101)Qn. Then of course (cid:101)En,N GL3n N We now de(cid:27)ne gauge transformations on the networks (cid:101)Qn. At any vertex, we may choose some A ∈ GLN, and multiply all incoming edge weights at that vertex by A on the right, and all outgoing edge weights by A−1 on the left. Clearly, the weight of any path passing through this vertex is unchanged by this action. We let G be the group generated by these transformations. We would like to think of the gauge group acting on the space (cid:101)En,N. However, this space consists only of those assignments of weights in which all edges other than the left, right, and bottom of each square face have weight equal to IdN. The gauge transformations described above do not, in general, preserve this property. So we will instead restrict to the subgroup G◦ ⊂ G 42 which preserves(cid:101)En,N. We will now describe a system of generators for G◦. The top edge of each square face has six “nearby” nontrivial weights – three to the left, and three to the right. For a matrix X ∈ GLN, the corresponding generator of G◦ will multiply the three weights to the left by X on the right, and multiply the three weights to the right by X−1 on the left. This is pictured in Figure 5.1. Figure 5.1: A generator of the gauge subgroup G◦ Ai−2 Ci−1 Bi−1 Ai−1 Bi+1 Ai+1 Ci Ai Bi Bi+2 Ci+1 Ci+2 (cid:32) Ai−2 Ci−1X Bi+1 X−1Ai+1 Ci+2 X−1Bi+2 Bi−1X Ai−1X X−1Ci Bi Ai Ci+1 Also, the subgroup G• ⊂ G◦ consisting of gauge transformations which are n-periodic can be considered to act on En,N, thought of as the space of weights for (cid:98)Qn on the torus, since En,N is naturally identi(cid:27)ed with the subset of (cid:101)En,N with n-periodic weights. We will now consider several quotients of T Ln,N by di(cid:29)erent group actions. First, we con- sider the action of R∗ on T Ln,N by dilations. For λ ∈ R∗, the action is given by Vi (cid:55)→ λVi, and the monodromy matrix is unchanged. This obviously does not change the polygon which the lift represents. We let(cid:103)T Ln,N denote the quotient by this action. Since the action is free and proper, we have dim(cid:103)T Ln,N = dimT Ln,N − 1 = 3N 2(n + 3) − 1 Now we may de(cid:27)ne an action of PGL3N on (cid:103)T Ln,N. Let A ∈ GL3N be a representative of A ∈ PGL3N, and let (V1, . . . , Vn, M ) ∈ T Ln,N. Then A acts in the usual way by Vi (cid:55)→ AVi and M (cid:55)→ AM A−1. The action by conjugation is well-de(cid:27)ned on PGL3N, since scalar matrices act trivially. The left action Vi (cid:55)→ AVi is also well-de(cid:27)ned, since A = λA and Vi = λ−1Vi. This 43 action is also free and proper, so dim(cid:103)T Ln,N /PGL3N = dim(cid:103)T Ln,N − dim PGL3N = 3nN 2 Theorem 5. The spaces T Ln,N and En,N are related in the following ways: (a) There is a bijective map(cid:103)T Ln,N /PGL3N → En,N (b) There is a bijective map (cid:101)Ln,N /PGL3N → (cid:101)En,N . (c) Under these identi(cid:27)cations, the actions of G◦ on (cid:101)En,N and G• on En,N correspond to changing the lift of a (cid:27)xed polygon. ∼= En,N /G• ∼= (cid:101)En,N /G◦. (d) GPn,N Proof. (a) Given a twisted lift (Vi) with monodromy M, we have for each i matrices Ai, Bi, and Ci in GLN so that Vi+3 = ViAi + Vi+1Bi + Vi+2Ci (5.2) The fact that the lift is twisted guarantees that the Ai, Bi, and Ci are periodic. Placing Ai−1 on the left edge of each square face, Bi on the bottom, and Ci on the right de(cid:27)nes a point in En,N. This gives a map T Ln,N → En,N. It is clear that two equivalent polygons have the same Ai, Bi, Ci coe(cid:28)cients for both the R∗ and the PGL3N actions. Therefore the proposed map(cid:103)T Ln,N /PGL3N → En,N is well-de(cid:27)ned. Now, supposing that V and W in(cid:103)T Ln,N have the same Ai, Bi, Ci coe(cid:28)cients, we wish to show that there is some g ∈ PGL3N so that W = gV . This will show that the map is injective. We 44 (cid:27)rst de(cid:27)ne the “transfer matrices” Li: Li :=  0 0 Ai IdN 0 Bi 0 IdN Ci  Now, if we form the block-column matrices Vi, whose columns are the combined columns of Vi, Vi+1, and Vi+2, then Equation 5.2 can be written succinctly as Vi+1 = ViLi. By the nondegeneracy assumption on lifts, det(Vi) (cid:54)= 0 for all i. For any two lifts V and W (even of two di(cid:29)erent polygons), there are matrices gi ∈ GL3N so that Wi = giVi for all i. We claim that if V and W have the same Ai,Bi,Ci coe(cid:28)cients, then all gi are the same. To see this, note that V and W will have the same transfer matrices Li. That is, Vi+1 = ViLi and Wi+1 = WiLi. Let Gi be the block diagonal matrices with gi, gi+1, gi+2 as the diagonal blocks. Then the equations Wi = giVi can be written as Wi = GiVi. Then we have Wi+1 = WiLi = GiViLi = GiVi+1 But also, by de(cid:27)nition of the Gi, we have Wi+1 = Gi+1Vi+1. So it must be that gi+1 = gi for all i. We have shown that for two lifts V and W in the same (cid:27)ber of the map T Ln,N → En,N, there is some g ∈ GL3N so that W = gV . In terms of the elements of(cid:103)T Ln,N, we can say that W = gV for the corresponding element g ∈ PGL3N. This shows the map is injective. To see that the map is surjective, let Ai, Bi, Ci be an arbitrary choice of weights in En,N. We will construct a lift which maps to this choice of weights. We will de(cid:27)ne V1, V2, and V3 so that the matrix V1 = (V1V2V3) is the identity matrix Id3N. We then de(cid:27)ne the rest of the Vi by Equation 5.2. Any other equivalent polygon is given by a di(cid:29)erent choice of V1, which di(cid:29)ers 45 only by the PGL3N action, giving the same point in(cid:103)T Ln,N /PGL3N. This gives an inverse to (b) There is the obvious map Ln,N → (cid:101)En,N which sends a lifted polygon V to the Ai, Bi, Ci the map described above. de(cid:27)ned above. If the lift is not twisted, these are not necessarily periodic. Two lifted polygons V and W have the same Ai, Bi, Ci coe(cid:28)cients if and only if Wi = gVi for all i and some (cid:27)xed g ∈ GL3N, as described above. The proof is the same as in part (a). This shows that the induced map (cid:101)Ln,N /PGL3N → (cid:101)En,N is bijective. (c) To see the equivalence of the gauge action of G◦ and changing lifts of a polygon, we will use the following visualization technique. We will associate to the top edge of each square face the lifted vertex Vi. The left edge will be labeled by Ai−3, the bottom with Bi−2, and the right by Ci−2. Then the relation Vi+3 = ViBi + Vi+1Ai + Vi+2Ci can be visualized in the following way. We sum over all paths from Vj to Vi which do not pass over another Vk in between. We see that there are only 3 such paths, which start at Vi, Vi+1, and Vi+2. The weights of the paths are the matrix coe(cid:28)cients which we multiply by on the right. The claim is that the action of the generators of G◦ and G• described above correspond pre- cisely to changing to a di(cid:29)erent lift of the same twisted polygon. So consider changing one lifted vertex Vi to (cid:98)Vi = ViG for some G ∈ GLN. Each Vi is involved in exactly four versions of Equation 5.2. Since Vi = (cid:98)ViG−1, these four relations for the new lift become: Vi+3 = (cid:98)Vi(G−1Bi) + Vi+1Ai + Vi+2Ci Vi+2 = Vi−1Bi−1 +(cid:98)Vi(G−1Ai−1) + Vi+1Ci−1 Vi+1 = Vi−2Bi−2 + Vi−1Ai−2 +(cid:98)Vi(G−1Ci−2) (cid:98)Vi = Vi−3(Bi−3G) + Vi−2(Ai−3G) + Vi−1(Ci−3G) 46 These are precisely how the weights change in Figure 5.1 by taking X = G. (d) Recall that GPn,N is the quotient of T Ln,N by the actions of GL3N and of changing the lift. By part (a), we identify En,N with the space of twisted lifts, up to the actions of R∗ and PGL3N. The R∗ action is one way of changing the lift, and corresponds to a subgroup of G•. This subgroup is the diagonal embedding of the scalar matrices {λIdN} into GLn . By part (c), the action of G• on En,N is equivalent to changing the lift. We may thus identify GPn,N with En,N /G•. Similarly, we may describe it also as GPn,N ∼= (cid:101)En,N /G◦. N We now use the gauge action to choose a convenient normalization, which we will use to coordinatize the moduli space. The following is an analogue of Proposition 1. Proposition 6. The lift (Vi) of (pi) can be chosen so that Ci = IdN for all i. Proof. We start by choosing a twisted lift Vi. This guarantees that the Ai, Bi, and Ci are periodic. Any other lift (cid:98)Vi di(cid:29)ers from Vi by a change of basis. That is, there are Gi ∈ GLN so that (cid:98)Vi = ViGi. Equivalently, Vi = (cid:98)ViG−1 (cid:17) . Substituting this into Equation 5.1, we obtain (cid:16) (cid:17) (cid:17) i (cid:16) +(cid:98)Vi+1 (cid:16) +(cid:98)Vi+2 (cid:98)Vi+3 = (cid:98)Vi G−1 i AiGi+3 G−1 i+1BiGi+3 G−1 i+2CiGi+3 i Gi+2. Re-indexing gives Gi+1 = C−1 We want to prove that we can always ensure that G−1 i+2CiGi+3 = Id. In other words, we desire Gi+3 = C−1 i−2Gi. Now we may choose for instance G0 = Id, and determine the rest by this recurrence. Equivalently, we may think of this in terms of the weights in (cid:101)En,N and the gauge action of G◦. Solving the recurrence above means (cid:27)rst choosing a position (a square face) in the network, and then applying elementary gauge transformations to the left and right to cancel the weights on the right edges of each square face. 47 The lift obtained in the above proposition is identi(cid:27)ed with a choice of weights in (cid:101)En,N for which the only non-trivial weights are on the left and bottom edges of each square face. We call the weights on the left edges Xi, and the ones on the bottom Yi, so that Equation 5.1 becomes Vi+3 = ViYi + Vi+1Xi + Vi+2 (5.3) Remark. It is worth pointing out that although in the proof of the above proposition, we started with a twisted lift, the resulting weights in(cid:101)En,N are not periodic, meaning the lift guaranteed by the proposition is not twisted. However, the Xi, Yi weights are almost periodic in the following sense. De(cid:27)ne the matrix which is the cyclic product of the Ci’s: Z := Cn−1CnC1 ··· Cn−2 Then shifting the indices by n corresponds to conjugation by Z. That is, Xi+n = Z−1 Xi Z Yi+n = Z−1 Yi Z So although the weights themselves are not periodic, their conjugacy classes are periodic. As a consequence of this normalization, we have a new model for the moduli space. Theorem 6. There is a bijection between the moduli space GPn,N and GL2n+1 /Ad GLN , the space of (2n+1)-tuples of matrices up to simultaneous conjugation. Therefore we have that dimGPn,N = 2nN 2 + 1. N Proof. We would like to say that the collection of matrices Xi, Yi, Z for 1 ≤ i ≤ n de(cid:27)nes a 48 N . However, this is not quite well-de(cid:27)ned. This depended on our choice map GPn,N → GL2n+1 of lift. The gauge action does not always preserve this collection of Xi, Yi, Z matrices, but in general changes them all by simultaneous conjugation. Therefore the induced map GPn,N → GL2n+1 /Ad GLN is well-de(cid:27)ned. N The map is injective, since the Xi’s, Yi’s and Z determine the class of a polygon. To see this, choose any elements for V1, V2, and V3. We may for instance choose them so that V1 = Id3N. Then the rest of the Vi are determined by the relations Vi+3 = ViYi + Vi+1Xi + Vi+2. However, the Xi’s, Yi’s and Z are only de(cid:27)ned up to simultaneous conjugation. Choosing some matrix A and conjugating all the Xi, Yi and Z by A, however, changes the lift by Vi (cid:55)→ ViA for all i. This just changes to a di(cid:29)erent lift of the same polygon, and so the choice of representatives of the Xi, Yi, and Z do not matter. Also, any other choice of the initial V1, V2, V3 di(cid:29)ers by the left action of PGL3N, and so gives an equivalent polygon. This gives an inverse to the map GPn,N → GL2n+1 /Ad GLN. N N Remark. The space GL2n+1 /Ad GLN parameterizes isomorphism classes of N-dimensional representations of the free associative algebra on 2n + 1 generators. Later, we will think of the matrices Xi, Yi, and Z as formal non-commutative variables (generators of a free algebra), and we will identify our moduli space GPn,N with the moduli space of N-dimensional representations of this algebra. With this description of the moduli space, we can now give some convenient coordinates. There is the following classical theorem of Procesi: Theorem 7. [Pro76] Any polynomial invariant of an n-tuple of matrices (A1, . . . , An), under the action of simultaneous conjugation, is a polynomial in the functions tr(Ai1 ··· Aik ), where ··· Aik Ai1 ranges over all non-commutative monomials in the matrices A1, . . . , An. 49 As mentioned above, we identify GPn,N with the quotient GL2n+1 N /Ad GLN. We consider . Procesi’s the coordinate ring to be the subring of invariants of polynomial functions on GL2n+1 theorem says that this coordinate ring is generated by the traces of monomials in 2n+1 matrices. As coordinates for the moduli space GPn,N, we may therefore take any algebraically independent family of traces of size 2nN 2 + 1. N 2/Ad GL2, the space of triples (X, Y, Z) of Example. (n = 1, N = 2) Consider the case of GL3 2 × 2 matrices up to simultaneous conjugation. A theorem of Sibiirski [Sib68] says that the ring of invariants is minimally generated by the ten functions tr(X) tr(Y ) tr(Z) tr(X2) tr(Y 2) tr(Z2) tr(XY ) tr(XZ) tr(Y Z) tr(XY Z) However, the dimension of the space is 9. We claim that the (cid:27)rst 9 (all but tr(XY Z)) can be taken . The as a system of local coordinates. The tangent space to GL3 2 gradients of the ten functions above are given by may be identi(cid:27)ed with Mat3 2 (Id2, 0, 0) (XT , 0, 0) (0, Id2, 0) (0, Y T , 0) (0, 0, Id2) (0, 0, ZT ) (Y T , XT , 0) (ZT , 0, XT ) (0, ZT , Y T ) ((Y Z)T , (ZX)T , (XY )T ) The (cid:27)rst 9 are clearly linearly independent for generic choices of X, Y , and Z. The fact that the last is a linear combination of the (cid:27)rst 9 is equivalent to the statement that there are coe(cid:28)cients 50 c1, . . . , c9 for which ZY = c1Id2 + c4X + c7Y + c8Z XZ = c2Id2 + c7X + c5Y + c9Z Y X = c3Id2 + c8X + c9Y + c6Z Of course, if X, Y , and Z are linearly independent, then the set {Id2, X, Y, Z} is a linear basis of the set of 2 × 2 matrices, and we can expand the products Y X, ZY , and XZ in terms of this basis. The claim above is that when we do this, three pairs of coe(cid:28)cients will be equal — the X- coe(cid:28)cient of Y X is equal to the Z-coe(cid:28)cient of ZY , etc. The fact that these pairs of coe(cid:28)cients will be equal follows from the Jacobi identity: [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0 5.4 Expression for the Pentagram Map In this section, we see how the pentagram map transforms the Xi, Yi, and Z matrices. The map is only de(cid:27)ned generically, since we need to assume certain matrices are invertible. Proposition 7. The pentagram map transforms the Xi, Yi, and Z by Xi (cid:55)→ (Xi + Yi+1)−1Xi(Xi+2 + Yi+3) Yi (cid:55)→ (Xi + Yi+1)−1Yi+1(Xi+2 + Yi+3) Z (cid:55)→ Z 51 Proof. The proof is essentially the same as that of Proposition 2. By re-arranging Equation 5.3, we see that for a lift of the image, we may take for T (Vi) the subspace spanned by the columns of Vi+3 − Vi+1Xi = ViYi + Vi+2 (5.4) Since the lift of the image under the pentagram map is again a twisted polygon, it satis(cid:27)es Equation 5.1: T (Vi+3) = T (Vi)Ai + T (Vi+1)Bi + T (Vi+2)Ci We substitute the expressions from Equation 5.4 into the equation above, using either the left- hand or right-hand side, according to the same convention used in the proof of Proposition 2. Doing so, we obtain Vi+5 + Vi+3Yi+3 = Vi+1 (Yi+1Bi − XiAi) + Vi+3 (Ai + Bi − Xi+2Ci) + Vi+5Ci Comparing coe(cid:28)cients of each Vi, we conclude that Ci = Id, and that XiAi = Yi+1Bi Yi+3 + Xi+2 = Ai + Bi Assuming that each Xi + Yi+1 is invertible, these equations can be solved for Ai and Bi to give the desired result. To see that Z is unchanged, it is a simple check that a shift of indices by n still corresponds to conjugation by the same Z. Notice that the expressions in the proposition are a non-commutative version of the expres- 52 sions obtained in the classical case, in Proposition 2. For the remainder of the paper, we attempt to generalize much of what was done in the previous expository sections to non-commutative variables, using elements of a non-commutative ring in place of the commutative coordinates on EQ, and we introduce a non-commutative Poisson structure which mimics the classical counter- part in [GSTV16]. 53 Chapter 6 Non-Commutative Poisson Structures 6.1 Double Brackets In this section, we discuss the basic constructions and de(cid:27)nitions introduced by Van den Bergh related to double brackets [Ber08]. Throughout, K is a (cid:27)eld, and A is an associative K-algebra. Unadorned tensor products are assumed to be over K. We will assume the characteristic of K is zero, but we do not necessarily assume it is algebraically closed. If A is an associative algebra, then A⊗n is an A-bimodule in the obvious way: x · (a1 ⊗ a2 ⊗ ··· ⊗ an−1 ⊗ an) · y = xa1 ⊗ a2 ⊗ ··· ⊗ an−1 ⊗ any We refer to this as the outer bimodule structure on A⊗n. De(cid:27)nition 17. A double bracket on A is a K-bilinear map {{−,−}} : A × A → A ⊗ A which satis(cid:27)es: (1) {{−,−}} is a derivation in the second argument with respect to the outer bimodule structure: {{a, bc}} = {{a, b}} (1 ⊗ c) + (b ⊗ 1){{a, c}} 54 (2) {{b, a}} = −{{a, b}}τ, where (x ⊗ y)τ := y ⊗ x Note that these properties imply that {{−,−}} also satis(cid:27)es {{ab, c}} = (1 ⊗ a){{b, c}} + {{a, c}} (b ⊗ 1) De(cid:27)nition 18. A double bracket {{−,−}} is called a double Poisson bracket if it additionally sat- is(cid:27)es a version of the Jacobi identity: 2(cid:88) k=0 0 = σk ◦ ({{−,−}} ⊗ Id) ◦ (Id ⊗ {{−,−}}) ◦ σ−k The right-hand side is an operator on A⊗ A⊗ A, and σ is the permutation operator which sends x ⊗ y ⊗ z to z ⊗ x ⊗ y. For an algebra A, let µ : A ⊗ A → A be the multiplication map. If A has a double bracket {{−,−}}, then we de(cid:27)ne another operation {−,−} : A × A → A by composing with µ: {a, b} := µ({{a, b}}) Proposition 8. [Ber08] Suppose {{−,−}} is a double bracket on A (not necessarily a double Poisson bracket). The induced bracket {−,−} has the following properties 1. {a, bc} = {a, b}c + b{a, c} 2. {ab, c} = {ba, c} 3. {a, b} ≡ −{b, a} mod [A, A] Proof. Suppose that {{a, b}} =(cid:80) i ωi ⊗ ¯ωi and {{a, c}} =(cid:80) 55 (Leibniz in the 2nd argument) (cyclic in the 1st argument) (skew mod commutators) i ηi ⊗ ¯ηi. Then using the Leibniz rule for {{−,−}} in the second argument, and composing with multiplication, we get (cid:88) (cid:88) {a, bc} = ω ¯ωic + bηi ¯ηi = {a, b}c + b{a, c} i i This proves the (cid:27)rst identity. The third identity follows from the fact that For the second identity, suppose that {{b, c}} =(cid:80) i ζi ⊗ ¯ζi. Then on the one hand, we have {a, b} + {b, a} = (cid:88) [ωi, ¯ωi] i (cid:88) (cid:88) {{ab, c}} = ηib ⊗ ¯ηi + ζi ⊗ a¯ζi On the other hand, we have i i {{ba, c}} = (cid:88) i ηi ⊗ b¯ηi + (cid:88) i ζia ⊗ ¯ζi Obviously both will be the same after composing with the multiplication map. De(cid:27)nition 19. For an associative algebra A, de(cid:27)ne the cyclic space, denoted A(cid:92), to be the vector space quotient A/[A, A], by the linear span of all commutators. Note that since [A, A] is not in general an ideal, this is not necessarily an algebra. For a vector subspace V ≤ A, we let V (cid:92) denote its image under the projection. Similarly, for an element a ∈ A, we use the notation a(cid:92) to denote its image under the projection. Note that conjugate elements are equivalent, since xyx−1 − y = [xy, x−1]. From this it follows that in A(cid:92), monomials are equivalent up to cyclic permutation, since for any x1, . . . , xn ∈ A, we have xnx1 ··· xn−1 = xn(x1 ··· xn)x−1 n . 56 The following follows easily from the properties above. Proposition 9. [Ber08] Suppose {{−,−}} is a double bracket on A. Then there is a well-de(cid:27)ned bilinear skew-symmetric map (cid:104)−,−(cid:105) : A(cid:92) × A(cid:92) → A(cid:92), given by (cid:68) a(cid:92), b(cid:92)(cid:69) := {a, b}(cid:92) The next few results indicate the extra structure which {−,−} and (cid:104)−,−(cid:105) inherit if {{−,−}} additionally satis(cid:27)es the double Jacobi identity. Proposition 10. [Ber08] Suppose that {{−,−}} is a double Poisson bracket. Then 1. {−,−} is a “Loday bracket” on A. That is, it satis(cid:27)es a version of the Jacobi identity: {a,{b, c}} = {{a, b}, c} + {b,{a, c}} 2. (cid:104)−,−(cid:105) is a Lie bracket on A(cid:92). A more general notion is given by what William Crawley-Boevey calls an H0-Poisson struc- ture: De(cid:27)nition 20. [CB11] An H0-Poisson structure on an associative algebra A is a Lie bracket [−,−] on A(cid:92) such that each [a,−] is induced by a derivation of A. In particular, the induced bracket(cid:104)−,−(cid:105) of a double Poisson bracket{{−,−}} is an H0-Poisson structure by Proposition 8 and Proposition 10. However, there are H0-Poisson structures which do not arise from double Poisson brackets. Later, we will de(cid:27)ne a double bracket which is not double Poisson, but still has the property that the induced bracket (cid:104)−,−(cid:105) on A(cid:92) is a Lie 57 bracket. This will be an example of an H0-Poisson structure which comes from a double bracket, but not a double Poisson bracket. There is a theorem of Crawley-Boevey which says that H0-Poisson structures on A induce usual “commutative” Poisson structures on the representation space RepN (A) := Hom(A, MatN )/Ad GLN Theorem 8. [CB11] If A is an associative algebra, and the bracket (cid:104)−,−(cid:105) on A(cid:92) de(cid:27)nes an H0- Poisson structure, then there is a unique Poisson structure on RepN (A) such that the trace map is a Lie algebra homomorphism. That is, {tr(a), tr(b)} = tr (cid:68) a(cid:92), b(cid:92)(cid:69) Recall that by Theorem 6, our moduli space GPn,N is identi(cid:27)ed with GL2n+1 /Ad GLN. We will consider our algebra A to be a free algebra with 2n+1 generators, corresponding to the Xi, Yi, and Z. Since any homomorphism with domain A is determined by the images of the generators, we see that RepN (A) ∼= Mat2n+1 /Ad GLN. We may therefore interpret tr(Xi), tr(Yi), and tr(Z) as functions on GPn,N. We will spend the next couple sections de(cid:27)ning non-commutative Poisson structures on this free algebra. The theorem above will then induce a Poisson structure N N on our moduli space. 58 Chapter 7 Non-Commutative Networks and Double Brackets 7.1 Weighted Directed Fat Graphs (Revisited) Let Q = (Q0, Q1) be a quiver as in section 2, which is embedded in an annulus, with all sources on the inner boundary circle, all sinks on the outer boundary circle, and all internal vertices trivalent. De(cid:27)ne the algebra A = AQ = Q(cid:104)α | α ∈ Q1(cid:105) to be the free associative algebra generated by the arrows of the quiver. Assuming, as before, that Q is acyclic, we can de(cid:27)ne the boundary measurement matrix BQ(λ) = (bij(λ)), where bij(λ) is the sum of the weights of all paths from source i to sink j. Here, the weight of a path is the product of the weights, in order from left to right. We again assign an indeterminate λ to the cut, and consider the boundary measurements to be Laurent polynomials in A[λ±], where λ commutes with elements in A. We can de(cid:27)ne non-commutative gauge transformations. For any Laurent monomial t in the generators of A, we can multiply all incoming arrows at one vertex by t on the right, and multiply all outgoing arrows at that vertex by t−1 on the left. This will obviously not change the boundary measurements. We can also de(cid:27)ne non-commutative versions of the the Postnikov moves, which preserve the boundary measurements. They are pictured in Figure 7.1. As in the commutative case, 59 ∆ := b + adc. We must take special care at this point to say what we mean by expressions such as (b + acd)−1. We do so now. Figure 7.1: Non-commutative Postnikov moves ( I ) ( II ) ( III ) d b e4 c e3 e1 a e2 x a b c y e4 e1 dc∆−1bc−1 dc∆−1 ∆−1ad e2 ∆ e3 b−1x a b by c x xb a b c a b c y yb−1 7.2 The Free Skew Field and Mal’cev Neumann Series In order to make sense of expressions of the form (x + y)−1, we need an appropriate notion of “non-commutative rational functions”. This will be the free skew (cid:27)eld, which we will de(cid:27)ne below. Then, we will discuss how the free skew (cid:27)eld can be identi(cid:27)ed with a certain subset of non-commutative formal power series. For a set X = (x1, . . . , xn) of formal non-commuting variables, the free skew (cid:27)eld or universal (cid:27)eld of fractions, which we will denote by FQ(X), or just F(X), is a division algebra over Q 60 characterized by the following universal property: There is an injective homomorphism i : Q(cid:104)X(cid:105) (cid:44)→ F(X) from the free associative algebra on X into F(X) such that for any homomorphism ϕ : Q(cid:104)X(cid:105) → D into a division ring D, there is a unique subring Q(cid:104)X(cid:105) ⊂ Rϕ ⊂ F(X) and a homomorphism ψ : R → D with ϕ = ψ ◦ i, and such that if 0 (cid:54)= a ∈ R and ψ(a) (cid:54)= 0, then a−1 ∈ R. The explicit construction of F(X) is a bit complicated (see [GGRW02] or [Coh77] for details), but informally it consists of non-commutative rational expressions in the variables x1, . . . , xn, under some suitable notion of equivalence. For example, F(X) contains expressions such as (1 + x)−1 and w(x + y)−1z. Another way to embed the free algebra into a division ring is by so-called Mal’cev Neumann series (also called Hahn-Mal’cev-Neumann series). Suppose that the free group generated by X is given some order relation compatible with multiplication. This means that if f ≤ g, then hf ≤ hg for any h. Given an ordering of the free group, the ring of Mal’cev Neumann series is the subset of all formal series over the free group which have well-ordered support. This is a division ring which contains Q(cid:104)X(cid:105). Jacques Lewin proved [Lew74] that the free skew (cid:27)eld F(X) is isomorphic to the sub(cid:27)eld of the ring of Mal’cev Neumann series generated by the variables x1, . . . , xn. In the next section, we show an example of expanding an element of the free skew (cid:27)eld as a series. Since A = AQ = Q(cid:104)Q1(cid:105) is a free algebra, it is a subalgebra of F(Q1). So, when we consider the non-commutative Postnikov “square move”, we interpret the expressions (b + adc)−1 as ele- ments of the free skew (cid:27)eld F(Q1). If we choose an order relation compatible with multiplication, then we may also view these expressions as non-commutative Mal’cev Neumann series. Thus, from now on, given a quiver, we will work more generally with the skew (cid:27)eld F(Q1) rather than the free algebra Q(cid:104)Q1(cid:105). 61 7.3 An Example of a Series Expansion Consider the free skew (cid:27)eld on two variables x and y. We will write the element (x + y)−1 as a non-commutative power series. As mentioned above, we need to put an order on the free group generated by x and y to determine the ring of Mal’cev Neumann series. We will use the order induced by the Magnus embedding of the free group into the ring Z[[x, y]] of formal power series in non-commuting variables x and y [MKS76]. More speci(cid:27)cally, we embed the free group into Z[[x, y]] by the map α (cid:55)→ 1 + α and α−1 (cid:55)→ (cid:80) i≥0(−1)iαi where α is either x or y. This embeds the free group as a subgroup of the multi- plicative group of all power series with constant term 1. Choosing an order of the variables, say x < y, this determines a graded lexicographic order on monomials/words in Z[[x, y]]. Then we de(cid:27)ne an order on Z[[x, y]] where f < g if the coe(cid:28)cient of f at the (cid:27)rst place they disagree is smaller. This induces an order on the free group. We then de(cid:27)ne the ring of Mal’cev Neumann series to be those series which have well-ordered support with respect to this ordering of the free group. There are a couple di(cid:29)erent ways we could try to expand (x + y)−1 as a series. We could factor out either of the variables, and then expand as a geometric series. For example, we could (cid:27)rst factor out x to get (x + y)−1 = (x(1 + x−1y))−1 = (1 + x−1y)−1x−1 Then we expand (1 + x−1y)−1 as a geometric series to get (x + y)−1 = (1 − x−1y + x−1yx−1y + ··· )x−1 = x−1 − x−1yx−1 + x−1yx−1yx−1 + ··· 62 To see if the support is well-ordered, we need to go through the above construction. Under the Magnus embedding, we have x−1 (cid:55)→ 1 − x + x2 − x3 + ··· x−1yx−1 (cid:55)→ 1 − 2x + y + 3x2 − xy − yx + ··· x−1yx−1yx−1 (cid:55)→ 1 − 3x + 2y + 6x2 + y2 − 3xy − 3yx + ··· ... (x−1y)nx−1 (cid:55)→ 1 − (n + 1)x + ny + ··· The sequence of terms in our series expansion for (x + y)−1 all di(cid:29)er at the coe(cid:28)cient for x. We see that the coe(cid:28)cients are decreasing in the sequence −1,−2,−3,··· ,−(n + 1),··· . There is thus no lowest term in this sequence, and so it is not well-ordered. This expansion is then not a Mal’cev Neumann series for our chosen ordering. If, however, we factor out y instead of x, things will work out. Then we get (x + y)−1 = y−1(1 + xy−1)−1 = y−1 − y−1xy−1 + y−1xy−1xy−1 + ··· Then the general term, under the Magnus embedding, will be y−1(xy−1)n (cid:55)→ 1 + nx − (n + 1)y + ··· Now the sequence of x coe(cid:28)cients is the increasing sequence 0, 1, 2, 3,··· , n,··· . Therefore the (cid:27)rst term y−1 is the lowest in the sequence, and so the support is well-ordered. This expansion is thus a Mal’cev Neumann series. 63 7.4 Double Brackets Associated to a Quiver We now de(cid:27)ne a family of double brackets on the skew (cid:27)eld FQ := F(Q1) which generalize the Poisson structures on the space of edge weights EQ described in section 2. We again de(cid:27)ne it locally, and then describe the concatenation procedure by which they can be glued together. We start with the two local pictures of white and black vertices: z x y c a b Let F◦ = F(x, y, z) be the free skew (cid:27)eld (over Q) on 3 generators. We will think of the variables x, y, z as representing the edge weights on the white vertex picture above. Similarly, let F• = F(a, b, c) correspond to the black vertex. Of course they are isomorphic as associative Q- algebras, but we will de(cid:27)ne di(cid:29)erent brackets on them. Choose any w1, w2, w3 ∈ Q, and de(cid:27)ne a double bracket {{−,−}}◦ on F◦ by {{x, y}}◦ = w1(1 ⊗ xy) {{x, z}}◦ = w2(1 ⊗ xz) {{y, z}}◦ = w3(y ⊗ z) Similarly, for scalars k1, k2, k3 ∈ Q, de(cid:27)ne {{−,−}}• on F• by 64 {{a, b}}• = k1(ba ⊗ 1) {{a, c}}• = k2(ca ⊗ 1) {{b, c}}• = k3(c ⊗ b) Also de(cid:27)ne F∂ = F(x) = Q(x) to be the (cid:27)eld of rational functions in one variable, corre- sponding to a univalent boundary vertex, with trivial double bracket. These double brackets are again “compatible” with concatenating/glueing in a similar sense as in the commutative case. Let HQ be the free skew (cid:27)eld generated by the half-edges of Q, denoted again by αs and αt for the source and target ends of α ∈ Q1. Then HQ is the free product of the algebras de(cid:27)ned above: (cid:18) ∗◦∈V◦ (cid:19) F◦ ∗ (cid:18) ∗•∈V• ∼= HQ (cid:19) F• ∗ (cid:32) ∗ (cid:63)∈V∂ (cid:33) F∂ As is mentioned in [Ber08], for algebras A and B with double brackets, there is a uniquely de(cid:27)ned double bracket on the free product A∗B such that {{a, b}} = 0 for a ∈ A and b ∈ B. Since the free product is the coproduct in the category of associative algebras (analogously the tensor product is the coproduct for commutative algebras), this is analogous to the product bracket on HQ in the commutative case. Again we de(cid:27)ne a glueing map g∗ : FQ → HQ by g∗(α) = αsαt for α ∈ Q1. Since the algebras are non-commutative, this is not actually the pull-back of a geometric/topological map, but we keep the notation for the sake of analogy. We have the following result, which mimics the commutative case: 65 Proposition 11. There is a unique double bracket on FQ so that the glueing homomorphism g∗ : FQ → HQ satis(cid:27)es (cid:16){{α, β}}FQ (cid:17) {{g∗(α), g∗(β)}}HQ = (g∗ ⊗ g∗) Proof. As in the commutative case, there are many cases to consider, but all of them are similar and are simple calculations. We show one as an example to illustrate the idea. We again look at Figure 3.3, denoting by f the edge labeled ax, and try to de(cid:27)ne {{f, y}}FQ desired property, we must have . If it is to satisfy the (g∗ ⊗ g∗)({{f, y}}FQ ) = {{g∗(f ), g∗(y)}}HQ = {{ax, ysyt}}HQ (1 ⊗ yt) = (1 ⊗ a){{x, ys}}HQ = w1(1 ⊗ a)(1 ⊗ xys)(1 ⊗ yt) = w1(1 ⊗ axysyt) = w1(1 ⊗ g∗(f )g∗(y)) = (g∗ ⊗ g∗)(w1(1 ⊗ f y)) This suggests that {{f, y}}FQ must be de(cid:27)ned to be w1(1 ⊗ f y). As mentioned above, all other cases are similar. Just as in the commutative case, the double bracket on FQ is given by the same expressions as in HQ, treating edges as the corresponding half-edges which meet at a common vertex. We may also consider the doubled quiver Q, as before. We then associate to each opposite 66 arrow α∗ the element α−1 in the free skew (cid:27)eld. The double bracket de(cid:27)ned above on generators extends in a unique way to the free skew (cid:27)eld by the formulas β, α−1(cid:111)(cid:111) (cid:110)(cid:110) (cid:110)(cid:110) (cid:111)(cid:111) α−1, β = −(α−1 ⊗ 1){{β, α}} (1 ⊗ α−1) = −(1 ⊗ α−1){{α, β}} (α−1 ⊗ 1) We may then interpret any path in Q as a non-commutative Laurent monomial in FQ. 7.5 A Formula for the Bracket We will primarily be concerned with paths that are closed loops in Q. Let L ⊂ FQ be the vector subspace spanned by all monomials which represent closed loops, and let f, g ∈ L be two elements. As before, we let f ∩ g denote the set of all maximal common subpaths. We will prove that, as in the commutative case, we only need to consider the ends of common subpaths in order to compute (cid:104)f, g(cid:105). First we give a result about the local structure of the double bracket. Lemma 4. Let x and y be edges in Q. That is, x and y can either be arrows in the quiver, or “reverse” arrows. Then (a) If y follows x (i.e. s(y) = t(x)), then {{x, y}} = λ(1 ⊗ xy) for some λ ∈ Q. (b) If x and y have the same source, then {{x, y}} = λ(x ⊗ y) for some λ ∈ Q. If x and y have the same target, then {{x, y}} = λ(y ⊗ x) for some λ ∈ Q. (c) Proof. The pictures below show the possibilities at a white vertex. The red are for part (a), the blue for part (b), and green for part (c). Note that there are 3 additional possibilities for part (a), given by the reversal/inverse of the pairs shown in red. 67 As in the picture from the previous section, call the unique incoming arrow x, and the outgoing arrows y and z, in counter-clockwise order from x. Then the three pairs of edges in the picture above for (a) give (cid:110)(cid:110) (cid:111)(cid:111) {{x, z}} = w2(1 ⊗ xz) z−1, y (cid:110)(cid:110) y−1, x−1(cid:111)(cid:111) = w3(1 ⊗ z−1)(z ⊗ y)(z−1 ⊗ 1) = w3(1 ⊗ z−1y) = −w1(x−1 ⊗ y−1)(xy ⊗ 1)(y−1 ⊗ x−1) = −w1(1 ⊗ y−1x−1) For part (b), the pictured pairs give (cid:111)(cid:111) (cid:110)(cid:110) {{y, z}} = w3(y ⊗ z) x−1, y (cid:111)(cid:111) (cid:110)(cid:110) x−1, z = −w1(1 ⊗ x−1)(1 ⊗ xy)(x−1 ⊗ 1) = −w1(x−1 ⊗ y) = −w2(1 ⊗ x−1)(1 ⊗ xz)(x−1 ⊗ 1) = −w2(x−1 ⊗ z) For part (c), the pairs pictured above give x, y−1(cid:111)(cid:111) (cid:110)(cid:110) (cid:110)(cid:110) x, z−1(cid:111)(cid:111) (cid:110)(cid:110) y−1, z−1(cid:111)(cid:111) = −w1(y−1 ⊗ 1)(1 ⊗ xy)(1 ⊗ y−1) = −w1(y−1 ⊗ x) = −w2(z−1 ⊗ 1)(1 ⊗ xz)(1 ⊗ z−1) = −w2(z−1 ⊗ x) = w3(z−1 ⊗ y−1)(y ⊗ z)(y−1 ⊗ z−1) = w3(z−1 ⊗ y−1) All other possibilities at a black vertex are veri(cid:27)ed by similar calculations. 68 Lemma 5. Let f, g ∈ L . The induced bracket (cid:104)f, g(cid:105) depends only on the endpoints of maximal common subpaths of f and g. Proof. The result can be stated more technically as follows. Suppose that p is an edge in f and q is an edge in g. Expanding {{f, g}} with the Leibniz rule, there will be a term involving {{p, q}}. The result says that unless p and q are incident to a common vertex which is an endpoint of a maximal common subpath, then this term is either zero, or it cancels with another term in the Leibniz expansion. First of all, if p and q are not incident to the same vertex, then {{p, q}} = 0. Since each vertex is trivalent, if two paths go through a common vertex, then they must have at least one edge in common at that vertex. So every non-zero contribution {{p, q}} comes from when p and q belong to a common subpath of f and g. We now argue that even in this case, {{p, q}} is either zero or cancels unless p and q occur at the ends of a common subpath. For the remainder of the proof, let w be a maximal common subpath of f and g. We (cid:27)rst consider the simplest case, which is when p and q are consecutive edges in w. So suppose that q immediately follows p in w. Then we can write w = w1pqw2, and f and g can be written as f = f0w1pqw2 and g = g0w1pqw2. Then after expanding using the Leibniz rules, we get two terms in {{f, g}} involving {{p, q}} and {{q, p}}. By the previous lemma, {{p, q}} = λ(1 ⊗ pq) for some λ, and {{q, p}} = −λ(pq ⊗ 1). Then these two terms are given by λ(g0w1p ⊗ f0w1)(1 ⊗ pq)(qw2 ⊗ w2) = λg0w1pqw2 ⊗ f0w1pqw2 and 69 −λ(g0w1 ⊗ f0w1p)(pq ⊗ 1)(w2 ⊗ qw2) = −λg0w1pqw2 ⊗ f0w1pqw2 Obviously, these terms cancel. There is also the case that the path w intersects itself, so it may be possible that p and q share a common vertex, but do not occur consecutively in the path. There are three cases, corresponding to parts (a), (b), and (c) in the previous lemma. First we consider part (a) of the previous lemma. So suppose p and q are consecutive arrows in the quiver, but they occur non-consecutively in the path. Let r be the third edge incident to the common vertex of p and q, oriented outward. There are three cases: 1. p and q occur consecutively twice, so w = w1pqw2pqw3. 2. The path follows r after p, and p before q, so w = w1prw2pqw3. 3. The path follows r after p, and r−1 before q, so w = w1prw2r−1qw3. In cases (2) and (3), the vertex in question is the end of another common subpath, and so we don’t consider these cases. In case (1), the brackets of the (cid:27)rst p and (cid:27)rst q cancel by the previous discussion, since they are consecutive edges in the path. The terms coming from the bracket of the (cid:27)rst p with the second q and vice versa are given by λ(g0w1pqw2pqw2pqw3 ⊗ f0w1pqw3) and − λ(g0w1pqw2pqw2pqw3 ⊗ f0w1pqw3) All other cases correspond to cases (b) and (c) from the previous lemma. All these cases imply that the vertex which p and q share is the end of some other common subpath, and so we don’t need to consider these. We have thus covered all cases, and the lemma is proved. Write f and g as products of edges: f = f1 ··· fn and g = g1 ··· gm. Then by the Leibniz 70 rule: {{f, g}} = (cid:88) i,j (g1 ··· gj−1 ⊗ f1 ··· fi−1)(cid:8)(cid:8)fi, gj (cid:9)(cid:9) (fi+1 ··· fn ⊗ gj+1 ··· gm) The result of the preceding lemma says that this sum is only over the pairs fi, gj which are incident to a common vertex which is the end of a maximal common subpath of f and g. So we will now compute what happens at these vertices. As in the commutative case, there appear to be 36 cases to consider, since the two endpoints of a common subpath could each have one of two colors and one of three orientations. It happens again, however, that the orientations do not a(cid:29)ect the outcome, which we formulate now as a lemma. Lemma 6. The contribution to (cid:104)f, g(cid:105) coming from a common subpath w ∈ f ∩ g depends only on the colors of the endpoint vertices, and not on the orientations. Proof. Let us (cid:27)rst consider the beginning of a common subpath w, where the paths f and g (cid:27)rst come together. Then we may write f = f1 ··· fiw1 ··· fn and g = g1 ··· gjw1 ··· gm, where w1 = fi+1 = gj+1 is the (cid:27)rst edge in the common subpath w, and fi and gj are incident to the same vertex as w1. There are three terms in the expansion of {{f, g}} coming from this vertex, corresponding to w, given by (g1 ··· gj−1 ⊗ f1 ··· fi−1)(cid:8)(cid:8)fi, gj (cid:9)(cid:9) (w1fi+2 ··· fn ⊗ w1gj+2 ··· gm) (g1 ··· gj ⊗ f1 ··· fi−1){{fi, w1}} (w1fi+2 ··· fn ⊗ gj+2 ··· gm) (g1 ··· gj−1 ⊗ f1 ··· fi)(cid:8)(cid:8)w1, gj (cid:9)(cid:9) (fi+2 ··· fn ⊗ w1gj+2 ··· gm) 71 By Lemma 4, all three of these terms give the simple tensor g1 ··· gjw1fi+2 ··· fn ⊗ f1 ··· fiw1gj+2 ··· gm, but with di(cid:29)erent coe(cid:28)cients. Which coe(cid:28)cient goes with which term depends on the orientation of the vertex. Figure 7.2 illustrates the three possibilities at a black vertex, with fiw1 in blue and gjw1 in red. Figure 7.2: Orientations of the beginning of a common subpath (i) (ii) (iii) The coe(cid:28)cients for the three pictures above are given in Table 7.1. In all three cases, the (cid:8)(cid:8)fi, gj (cid:8)(cid:8)w1, gj Table 7.1: Contributions to {{f, g}} from Figure 7.2 (cid:9)(cid:9) (cid:9)(cid:9) {{fi, w1}} Case k3(gj ⊗ fi) −k1(1 ⊗ fiw1) k2(gjw1 ⊗ 1) (i) k2(gj ⊗ fi) k3(1 ⊗ fiw1) −k1(gjw1 ⊗ 1) (ii) k3(gjw1 ⊗ 1) k2(1 ⊗ fiw1) (iii) −k1(gj ⊗ fi) three terms are the same simple tensor, so the coe(cid:28)cients add up. As we can see from the table, the combined coe(cid:28)cient is always −A• = k2 + k3 − k1. Similarly, for all three orientations at a white vertex, we will get the same simple tensor with a coe(cid:28)cient of A◦ = w1 − w2 − w3. If we consider the endpoint vertex of the common subpath, we will again get the same simple tensor for all three terms. If the vertex is black, we will get a coe(cid:28)cient of A•, and if it is white, we will get −A◦. These are exactly the same as the values for εw(f, g) in the commutative case. Let fwgw denote the loop which follows f starting with the path w, followed by g starting at 72 w. Then putting the previous lemmas together gives Theorem 9. Let f, g ∈ L . Then the induced bracket in F (cid:92) Q is given by (cid:88) x∈f∩g (cid:104)f, g(cid:105) = εx(f, g)fxgx In particular, L (cid:92) is closed under (cid:104)−,−(cid:105). Moreover, we have the following important observation. Theorem 10. The induced bracket (cid:104)−,−(cid:105) makes L (cid:92) a Lie algebra. Proof. We only need to verify the Jacobi identity. So let f, g, h ∈ L . Then (cid:104)f,(cid:104)g, h(cid:105)(cid:105) = = (cid:104)(cid:104)f, g(cid:105) , h(cid:105) = = (cid:88) (cid:88) i∈g∩h i∈g∩h (cid:88) (cid:88) j∈f∩g j∈f∩g εi(g, h)(cid:104)f, gihi(cid:105)  (cid:88) j∈f∩g  εk(f, h)fk(gihi)k (cid:88) k∈f∩h εj(f, g)fj(gihi)j + εi(g, h) εj(f, g)(cid:10)fjgj, h(cid:11)  (cid:88) εj(f, g) k∈f∩h εk(f, h)(fjgj)khk +  εi(g, h)(fjgj)ihi (cid:88) i∈g∩h 73 (cid:104)g,(cid:104)f, h(cid:105)(cid:105) = = (cid:88) (cid:88) k∈f∩h k∈f∩h εk(f, h)(cid:104)g, fkhk(cid:105)  (cid:88) i∈g∩h εk(f, h) εi(g, h)gi(fkhk)i − (cid:88) j∈f∩g  εj(f, g)gj(fkhk)j The Jacobi identity will hold if the (cid:27)rst equation is equal to the sum of the second and third. Comparing the right-hand side of the (cid:27)rst with the sum of the right-hand sides of the second and third, we see that the Jacobi identity will hold if the following identities are true: fj(gihi)j = (fjgj)ihi = gi(fjhj)i It is an easy check that these identities are indeed true for all cycles. This double bracket is not a double Poisson bracket (it does not satisfy the double Jacobi identity), so it does not immediately guarantee the Jacobi identity for (cid:104)−,−(cid:105). We do, however, have the Jacobi identity for (cid:104)−,−(cid:105) on the subset L by the previous theorem. Let • be a vertex in the quiver, and de(cid:27)ne L• to be the subspace spanned by loops based at •. This is clearly an associative subalgebra, since the (cid:27)xed basepoint allows us to concatenate paths. We also have the following. Proposition 12. The subalgebra L• is closed under the double bracket: {{L•, L•}} ⊆ L• ⊗ L• Proof. Let f, g ∈ L•. Write f and g as monomials, f = f1 ··· fk and g = g1 ··· g(cid:96), where each fi and gj are arrows in the quiver. Note that since they are both based at the point •, we have 74 s(f1) = s(g1) = t(fk) = t(g(cid:96)) = •. Now, using the Leibniz rule: (g1 ··· gj−1 ⊗ f1 ··· fi−1)(cid:8)(cid:8)fi, gj (cid:9)(cid:9) (fi+1 ··· fk ⊗ gj+1 ··· g(cid:96)) (cid:88) {{f, g}} = i,j Using the formulas from Lemma 4, and examining the three cases, we see that each term in the sum above is of the form α ⊗ β, where α and β are both in L•. This makes the induced bracket into an H0-Poisson structure on L•, as de(cid:27)ned by Crawley- Boevey [CB11]. 7.6 Goldman’s Bracket and the Twisted Ribbon Surface In this section, we give a geometric interpretation of the bracket just described. We (cid:27)rst recall some preliminaries. Let G be a connected Lie group and S a smooth oriented surface (with boundary) with fun- damental group π := π1(S). We consider the space of representations of π in G, modulo conju- gations, which we call RepG(π): RepG(π) := Hom(π, G)/G Let f : G → R be any invariant function on G, with respect to conjugation. Then for α ∈ π, we can de(cid:27)ne the function fα : RepG(π) → R by the formula fα([ϕ]) := f (ϕ(α)). In particular, if G is a group of matrices, we may take f = tr. In this case we will write tr(α) for fα. Also, if G = GLn(R), we will write Repn(π) instead of RepGLn(R)(π). In 1984, William Goldman described a symplectic structure on RepG(π) which generalizes the Weil-Petersson symplectic structure on Teichmüller space in the case that G = PSL2(R) [Gol84]. 75 Then, in 1986, he studied the Poisson bracket induced by this symplectic structure, in terms of the functions fα [Gol86]. Goldman gives explicit formulas for {fα, fβ} for various choices of the group G, in terms of the topology of the surface and the intersection of curves representing α and β. In particular, when G = GLn(R) and f = tr, we have the following. Theorem 11. [Gol86] The Poisson bracket of the functions tr(α) on Repn(π) is given by {tr(α), tr(β)} = εp(α, β) tr(αpβp) (cid:88) p∈α∩β Here, αpβp denotes the loop which traverses (cid:27)rst α, and then β, both based at the point p, and εp(α, β) is the oriented intersection number of the curves at p. Note the obvious similarity with our formula from Theorem 9. We will now formulate a geometric interpretation of our double bracket from a quiver so that a special choice of constants A• and A◦ realizes this Goldman bracket. First we de(cid:27)ne ˆπ to be the set of conjugacy classes in π (or free homotopy classes of loops). Goldman observes that the bracket above induces a Lie bracket on the vector space spanned by ˆπ. Formally, we simply remove the “tr” in the formula above. So for α, β ∈ ˆπ, we have (cid:88) p∈α∩β εp(α, β) αpβp [α, β] = With this setup, we have the following Theorem 12. [Gol86] Zˆπ is a Lie algebra with the bracket shown above, and the map tr : Zˆπ → C∞(Repn(π)) given by α (cid:55)→ tr(α) is a Lie algebra homomorphism. In fact, our proof of Theorem 10 is essentially the same as Goldman’s original proof ([Gol86], Theorem 5.3). 76 In the literature, fat graphs are also commonly called “ribbon graphs”. From a fat graph Γ, one can construct an oriented surface with boundary, SΓ, by replacing the edges with rectangular strips (ribbons) and the vertices with discs, where the ribbons are glued to the discs according to the cyclic ordering prescribed by the fat graph structure. We call this surface the ribbon surface associated to the fat graph. It is clear that the original graph Γ is a deformation retract of SΓ. We say that Γ is a spine of the surface S = SΓ. Given a quiver Q (oriented fat graph) with underlying unoriented graph Γ, we consider, as before, the subspace L ⊂ FQ of loops. Then in a natural way we identify the cyclic space L (cid:92) with Qˆπ, the space generated by free homotopy classes of loops on SΓ. In the commutative case, we considered the case A• = A◦ = − 1 , which gives εp(f, g) = 1 when f and g touch and εp(f, g) = 0 when f and g cross. We will again be primarily concerned with this speci(cid:27)c choice of coe(cid:28)cients in the non-commutative case. This is opposite from what εp(α, β) means for Goldman’s bracket, however. This is because if paths f and g touch in the quiver, then on SΓ, they are homologous to paths which do not touch at all, and so their Goldman bracket should be 2 zero. The way around this is to consider the “dual” surface. De(cid:27)nition 21. We de(cid:27)ne the dual surface to SΓ, which we denote(cid:101)SΓ, to be glued out of ribbons like SΓ, except that whenever an edge joins vertices of di(cid:29)erent colors, the corresponding ribbon is given a half-twist. Theorem 13. For the choice of coe(cid:28)cients A• = A◦ = − 1 2, the induced bracket (cid:104)−,−(cid:105) on L (cid:92) coincides with Goldman’s bracket under the identi(cid:27)cation of L (cid:92) with Qˆπ, where π = π1((cid:101)SΓ). Proof. We think of the quiver as a planar projection picture of the ribbon surface(cid:101)SΓ, where one of the surface (cid:101)SΓ, and try to view it without twists, we would see that paths which touch in the the colors is the “top” of the ribbon, and the other color is the “bottom”. If we were to “untwist” 77 quiver end up crossing in (cid:101)SΓ, and paths which cross in the quiver end up not touching in (cid:101)SΓ. Now that touching and crossing have been interchanged, we see that the Goldman bracket on(cid:101)SΓ coincides with the induced bracket (cid:104)−,−(cid:105) on L (cid:92) when A• = A◦ = − 1 2 . Recall that L• is the subalgebra of loops with basepoint •. Then L• is naturally identi(cid:27)ed with the group algebra of π = π1(SΓ,•). The induced bracket is independent of this choice of basepoint, since conjugate elements are equivalent in L (cid:92)• . We now point out some similar and related work which was brought to the author’s attention while working on this dissertation. Remark. In [MT12], Turaev and Massuyeau construct a quasi-Poisson bracket on the character variety Repn(π), which is induced by a double quasi-Poisson bracket on the group algebra of π. As was mentioned above, we may identify the group algebra of π with L•, and this also gives a double bracket on L•. However, the double bracket considered in this paper is not a double quasi- Poisson bracket, and so the two constructions are not exactly the same. However, the bracket of Turaev and Massuyeau also projects to the Goldman bracket in L (cid:92)• . Also, Semeon Artamonov, in his recent thesis [Art18], constructed a much more abstract and categorical version of the quasi-Poisson structure of Turaev and Massuyeau. 7.7 The X, Y Variables Just as in the commutative case, we will use gauge transformations to obtain new weights on the graph. Start with the quiver Qn just as in Figure 4.1. Give the edge weights the same names as in the commutative case, but now they are formal non-commutative variables in FQ. Per- form gauge transformations to obtain variables ai, bi, ci, di just as before. Note that the pictures in Section 4.2 are actually non-commutative gauge transformations, as they take into account 78 whether multiplication happens on the left or right. In this case, the double bracket induced on the a, b, c, d variables is given by: {{bi, ai}} = {{ai, di}} = bi ⊗ ai 1 ⊗ aidi 1 2 1 2 {{bi, ci}} = {{ci, di}} = ci ⊗ bi dici ⊗ 1 1 2 1 2 De(cid:27)ne the following monomials in the a, b, c, d variables: i−1d−1 i d−1 xi = aic−1 i−1c−1 i−2 yi = bic−1 i c−1 i−1d−1 i−1c−1 i−2 zk = d1c1 ··· dk−1ck−1dk These xi and yi are non-commutative versions of the same monomials given in the com- mutative case, and zk are paths connecting the upper-left corner of the (cid:27)rst square face to the upper-right corner of the kth square face. We may again perform the exact same sequence of gauge transformations as in the commutative case to arrive at the weights depicted in Figure 4.2. However, these weights will now be non-commutative versions of the same monomials. In terms of the monomials de(cid:27)ned above, the non-commutative weights we obtain in Figure 4.2 are given by 79 Xi = zi−2 xi z−1 i−2 Yi = zi−2 yi z−1 i−2 Z = zncn Just as in the commutative case, these can be interpreted as closed loops in the quiver. The di(cid:29)erence now in the non-commutative case is that all these loops share a common basepoint. This common basepoint is the upper-left corner of the (cid:27)rst square face. As in the previous section, let L• ⊂ L denote the subspace of loops based at this point. In fact, it is the group algebra of the fundamental group of the ribbon surface of the graph, and it is generated by the Xi, Yi, and Z. A simple calculation shows that Z is a Casimir of the double bracket. The bracket is then := L•/(cid:104)Z − 1(cid:105), where Z is set to 1. The induced brackets in well-de(cid:27)ned on the quotient L (1)• L (1)• are given by: (cid:104)Xi+1, Xi(cid:105) = Xi+1Xi (cid:104)Yi+2, Yi(cid:105) = Yi+2Yi (cid:104)Yi+1, Xi(cid:105) = Yi+1Xi (cid:104)Xi+2, Yi(cid:105) = Xi+2Yi (cid:104)Yi+1, Yi(cid:105) = Yi+1Yi (cid:104)Yi, Xi(cid:105) = YiXi (cid:104)Xi+1, Yi(cid:105) = Xi+1Yi If we don’t set Z = 1, then the bracket relations in L• are mostly the same, except some exceptions when i = 1 or i = 2: 80 (cid:104)X3, X2(cid:105) = X3Z−1X2Z (cid:104)Y3, X2(cid:105) = Y3Z−1X2Z (cid:104)Y3, Y1(cid:105) = Y3Z−1Y1Z (cid:104)X3, Y1(cid:105) = X3Z−1Y1Z (cid:104)Y3, Y2(cid:105) = Y3Z−1Y2Z (cid:104)X3, Y2(cid:105) = X3Z−1Y2Z (cid:104)Y4, Y2(cid:105) = Y4Z−1Y2Z (cid:104)X4, Y2(cid:105) = X4Z−1Y2Z If we perform the same sequence of Postnikov moves as in the commutative case, followed by gauge transformations, we get a graph isomorphism, with the same edges having weight 1 as before. This gives the transformation Xi (cid:55)→ (Xi + Yi)−1Xi(Xi+2 + Yi+2) Yi (cid:55)→ (Xi+1 + Yi+1)−1Yi+1(Xi+3 + Yi+3) Just as in the commutative case, this is almost the expression derived earlier for the pentagram map. It di(cid:29)ers only by a shift in the Y -indices. Remark. In the formula above, the indices are not read cyclically. If one of the indices is greater than n, we must conjugate by Z. For instance, if i = n − 2, and i + 3 = n + 1, then by Xi+3 we really mean ZX1Z−1, and similarly for Y . 81 7.8 The P, Q Variables We now de(cid:27)ne a non-commutative version of the face variables pi and qi from the classical case. They are given by pi = bic−1 i d−1 i a−1 i , qi = ci−2di−1ai+1b−1 i Conjugating by the same zk paths as for the X, Y variables, we obtain versions with a common basepoint: Pi = zi−2 pi z−1 i−2, Qi = zi−2 qi z−1 i−2 Similar to the commutative case, we have the following relation with the X, Y variables in the quotient L (1)• : Pi = YiX−1 i , Qi = Xi+1Y −1 i Note that this implies the relation QnPnQn−1Pn−1 ··· Q1P1 = 1. Their induced brackets are given by (cid:104)Qi, Pi(cid:105) = QiPi (cid:104)Qi+1, Pi(cid:105) = Qi+1Y −1 i PiYi (cid:104)Pi, Qi−1(cid:105) = PiQi−1 (cid:104)Pi, Qi+2(cid:105) = Y −1 i PiYiQi+2 From the formulas given in the previous section for the pentagram map in the X, Y variables, we get that the pentagram map transforms the P, Q variables by i+1)−1Xi+1 · (1 + Pi+3)Qi+2(1 + Pi+2) · Y −1 i Pi (cid:55)→ X−1 Qi (cid:55)→ X−1 i+1(1 + P−1 i+1P−1 i+1Xi+1 (1 + Pi)Yi 82 Chapter 8 Invariance, Invariants, and Integrability 8.1 Invariance of the Induced Bracket We will show in this section that the induced bracket (cid:104)−,−(cid:105) on L (cid:92)• is invariant under the pen- tagram map. To do so, we will consider step-by-step how the weights and double bracket change under the Postnikov moves. Recall that, starting with the X, Y variables, the sequence of Postnikov moves which gives the pentagram map is as follows: 1. Perform the square move at each square face 2. Perform the white-swap move at each edge connecting two white vertices 3. Perform the black-swap move at each edge connecting two black vertices 4. Perform gauge transformations so all weights are 1 except the bottom and left of each square face We will actually start with the a, b, c, d weights instead of the X, Y weights. The result will be the same, as we will explain below. Application of a square move gives 83 a d b c (cid:101)a (cid:101)d (cid:101)b (cid:101)c The new edge weights are given by (cid:101)b = b + adc (cid:101)c =(cid:101)b−1ad (cid:101)a = dc(cid:101)b−1 (cid:101)d =(cid:101)abc−1 The double brackets of these new weights are (cid:110)(cid:110)(cid:101)b,(cid:101)a (cid:111)(cid:111) (cid:110)(cid:110)(cid:101)b,(cid:101)c (cid:111)(cid:111) = = 1 2 1 2 (cid:16)(cid:101)a(cid:101)b ⊗ 1 (cid:17) (cid:16) (cid:17) 1 ⊗(cid:101)b(cid:101)c (cid:110)(cid:110)(cid:101)a,(cid:101)d (cid:111)(cid:111) (cid:110)(cid:110)(cid:101)c,(cid:101)d (cid:111)(cid:111) = = 1 2 1 2 (cid:16)(cid:101)a ⊗(cid:101)d (cid:17) (cid:16)(cid:101)d ⊗(cid:101)c (cid:17) Next, applying the white-swap and black-swap moves interchanges the square and octagonal faces. The resulting weights around the square faces is pictured in Figure 8.1. Figure 8.1: Edge weights after white-swap and black-swap moves (cid:101)ci−2 (cid:101)bi (cid:101)di−1 (cid:101)ai+1 Finally, we perform gauge transformations, as we did to get the original X, Y weights, so that all weights become 1 except those on the bottom and left edges of the square faces. We call these 84 resulting weights (cid:101)X and (cid:101)Y . If we de(cid:27)ne the “staircase” monomials ξk =(cid:101)a1(cid:101)b1 ···(cid:101)ak(cid:101)bk, then we can express the new weights as i+2(cid:101)b−1 (cid:101)Xi = ξi(cid:101)ci(cid:101)a−1 i+1(cid:101)a−1 i+1 ξ−1 i i+2(cid:101)b−1 i+3(cid:101)b−1 (cid:101)Yi = ξi(cid:101)di+1(cid:101)a−1 i+2(cid:101)a−1 i+1(cid:101)a−1 i+1 ξ−1 i These weights are the images of Xi and Yi under the pentagram map. That is, (cid:101)Xi = T (Xi) and(cid:101)Yi = T (Yi). As noted in the previous sections, these are given in terms of the original X, Y variables as (cid:101)Xi = σ−1 i+1Yi+1σi+3, where σi = Xi + Yi. We are now ready i Xiσi+2 and(cid:101)Yi = σ−1 to prove the main theorem of this section, but (cid:27)rst we introduce some notation. De(cid:27)ne the map S : L• → L• which shifts all the indices. That is, S(Xi) = Xi−1 and S(Yi) = Yi−1, where the indices are read cyclically. Theorem 14. In L (1)• (a) In L•, the induced bracket is invariant under S2 ◦ T . (b) , the induced bracket (cid:104)−,−(cid:105) is invariant under the pentagram map. Proof. We want to show that the induced brackets of (cid:101)Xi and (cid:101)Yi have exactly the same form as the induced brackets of the Xi and Yi. We will instead compute the brackets of the conjugate, but equivalent, elements: i+2(cid:101)b−1 (cid:101)Xiξi =(cid:101)ci(cid:101)a−1 i+1(cid:101)a−1 (cid:101)xi = ξ−1 We will compute the three possible combinations(cid:8)(cid:8)xi, xj i+2(cid:101)b−1 i+3(cid:101)b−1 i (cid:101)Yiξi = (cid:101)di+1(cid:101)a−1 i+1(cid:101)a−1 i+2(cid:101)a−1 (cid:9)(cid:9), and(cid:8)(cid:8)xi, yj (cid:9)(cid:9),(cid:8)(cid:8)yi, yj (cid:101)yi = ξ−1 i+1 i i+1 (cid:9)(cid:9). For 85 (cid:8)(cid:8)xi, xj (cid:9)(cid:9), a simple calculation using the Leibniz rules for double brackets gives: j (cid:101)xj +(cid:101)xj(cid:101)ai(cid:101)bi ⊗(cid:101)xi(cid:101)b−1 δi,j+1((cid:101)cj(cid:101)a−1 j+2(cid:101)xi ⊗(cid:101)aj+2(cid:101)c−1 i (cid:101)a−1 j ⊗(cid:101)xi(cid:101)aj(cid:101)bj +(cid:101)b−1 δi,j−1((cid:101)xj(cid:101)b−1 j ⊗(cid:101)xi(cid:101)aj(cid:101)bj ˜xj) j (cid:101)a−1 j (cid:101)a−1 (cid:8)(cid:8)(cid:101)xi,(cid:101)xj (cid:9)(cid:9) = i 1 2 − 1 2 ) Therefore their induced brackets are (cid:10)(cid:101)xi,(cid:101)xj (cid:11) = δi,j+1((cid:101)cj(cid:101)a−1 )(cid:101)xi ((cid:101)aj(cid:101)bj) ˜xj Conjugating the right hand side by ξj gives the equivalent formula in terms of the (cid:101)Xi: )(cid:101)xj − δi,j−1((cid:101)b−1 j (cid:101)a−1 j+2)(cid:101)xi ((cid:101)aj+2(cid:101)c−1 j j (cid:68)(cid:101)Xi, (cid:101)Xj (cid:69) = (δi,j+1 − δi,j−1)(cid:101)Xi(cid:101)Xj This is exactly the same as the bracket formula for the original Xi. There are, however, some exceptions. When i = n, we instead have (cid:69) (cid:68)(cid:101)X1, (cid:101)Xn (cid:69) and(cid:68)(cid:101)Xi,(cid:101)Yj = (cid:101)X1Z−1(cid:101)XnZ (cid:69) are similar, with exceptions when i = n and The calculations for(cid:68)(cid:101)Yi,(cid:101)Yj i = n− 1. So the exact form of the bracket relations are not invariant, since the exceptional cases (which are conjugated by Z) do not occur at the same indices. But if we set Z = 1, then the bracket is invariant in L (1)• . Also, if we shift the indices by 2 and de(cid:27)ne (cid:98)Xi := S2((cid:101)Xi) = S2 ◦ T (Xi), then the bracket is invariant under Xi (cid:55)→ (cid:98)Xi. Remark. Theorem 14 suggests that a more appropriate de(cid:27)nition of the pentagram map should include a “rotation”, which shifts the labels/indices of the vertices by 2. This shift of indices 86 is necessary to make the Poisson structure invariant. From now on, we will use the notation (cid:98)T : GPn,N → GPn,N for this modi(cid:27)ed version of the map. It is worth noting that earlier de(cid:27)- nitions of the pentagram map in [OST10] [Sch92] [GSTV16] do not include this shift of indices, and this phenomenon only appears by considering this non-commutative generalization. 8.2 The Invariants Let Q = Qn be the quiver/network for the pentagram map, with the Xi and Yi weights as before, and B = BQ(λ) its boundary measurement matrix, whose entries are elements of the Laurent polynomial ring L•[λ±]. For each i ≥ 1, denote the coe(cid:28)cients of λk in tr(Bi) by tik: (cid:88) k tr(Bi) = tikλk We will spend the remainder of this section proving the following theorem. Theorem 15. For each i and j, the classes t(cid:92) ij are invariant under both T and (cid:98)T . To begin proving this, we (cid:27)rst make the following simple observation: Lemma 7. Let R be an associative ring, and p, q ∈ R[λ±]. Then every coe(cid:28)cient of [p, q] is in i piλi and q =(cid:80) j qjλj. Then [p, q] =(cid:80) i,j[pi, qj]λi+j. [R, R]. Proof. Let p =(cid:80) Corollary 1. Let f = (cid:80) This implies the following i fiλi and g = (cid:80) j gjλj be Laurent polynomials in A := R[λ±]. If f ≡ g mod [A, A], then fi ≡ gi mod [R, R] for each i. We now consider traces of powers of matrices over general rings: 87 Lemma 8. Let R be an associative ring, and A, B ∈ Matn(R). Then tr((AB)k) ≡ tr((BA)k) mod [R, R] for all k. Proof. If the matrices are given by A = (aij) and B = (bij) then (AB)ij1 (AB)j1j2 (cid:88) (cid:88) (cid:88) i,j1,...,jk−1 i,j1,...,jk−1 (cid:88) (cid:88) (cid:96)1 tr(AB)k = = = ai(cid:96)1 b(cid:96)1j1 ajk−1(cid:96)k  b(cid:96)ki ··· (AB)jk−1i ··· (cid:88) (cid:96)k ··· ajk−1(cid:96)k b(cid:96)ki i,j1,...,jk−1 (cid:96)1,...,(cid:96)k ai(cid:96)1 b(cid:96)1j1 Similarly, using BA instead, we get tr(BA)k = (cid:88) (cid:88) i,j1,...,jk−1 (cid:96)1,...,(cid:96)k bi(cid:96)1 a(cid:96)1j1 ··· bjk−1(cid:96)k a(cid:96)ki After re-indexing by (cid:96)t (cid:55)→ jt (for 1 ≤ t ≤ k − 1), (cid:96)k (cid:55)→ i, i (cid:55)→ (cid:96)1, jt (cid:55)→ (cid:96)t+1, we get tr(BA)k = b(cid:96)1j1 aj1(cid:96)2 ··· b(cid:96)kiai(cid:96)1 (cid:88) (cid:88) i,j1,...,jk−1 (cid:96)1,...,(cid:96)k Clearly, this is the same as the expression for tr((AB)k) mod [R, R], since each term with corre- sponding indices is the same after cyclically shifting the last ai(cid:96)1 to the beginning. Now we may prove Theorem 15: Proof. As discussed earlier, the map T is given, in the X, Y variables, by a sequence of Postnikov moves and gauge transformations. The square move and gauge transformations do not change the boundary measurement matrix at all, and the “white-swap” and “black-swap” moves only 88 change the boundary measurement matrix up to conjugation, since it amounts to a refactorization B = B1B2 (cid:55)→ (cid:101)B = B2B1. If we let R = L• and A = L•[λ±], then by Corollary 1 and Lemma 8, tr( (cid:101)Bk) and tr(Bk) di(cid:29)er by an element of [A, A], and so t(cid:92) map (cid:98)T is just T followed by a shift of indices, which is equivalent to cutting part of the network is invariant under T. The ij o(cid:29) one end, and glueing it to the opposite end. Again, this amounts to a re-factorization of the boundary measurement measurement matrix, which changes it only up to conjugation. Remark. In [Izo18], Izosimov interprets the pentagram map, as well as some generalizations, in terms of re-factorizations in Poisson-Lie groups. At the end of the paper, he poses the question of whether his techniques, when applied to matrix-valued coe(cid:28)cients, give rise to the Grassmann pentagram map of Mari Be(cid:29)a and Felipe. It seems that the proof of Theorem 15, which realizes the pentagram map as a re-factorization of the boundary measurement matrix, suggests a positive answer to Izosimov’s question. 8.3 Involutivity of the Invariants In this section, we will prove that the invariants from the previous section are an involutive family with respect to the induced bracket. More speci(cid:27)cally: Theorem 16. Let Qn be the network for the pentagram map on GPn,N , and B = BQ(λ) its boundary measurement matrix, with tik the homogeneous components of tr(Bi) as de(cid:27)ned before. Then for all i, j, k, (cid:96): (cid:10)tik, tj(cid:96) (cid:11) = 0 Recall that we use as the cut the identi(cid:27)ed top/bottom edge of the rectangle on which we draw the quiver. So the element tik ∈ L is the sum over all loops in Q which are homologous 89 to (i, k) cycles on the torus. That is, if we lift these paths to the universal cover, then they cross fundamental domains i times horizontally and k times vertically (with sign). Let Aik be the set of all such paths, so that (cid:88) p∈Aik Then by bilinearity and the formula from Theorem 9, tik = p (cid:10)tik, tj(cid:96) (cid:11) = (cid:88) (cid:88) f∈Aik g∈Aj(cid:96) (cid:104)f, g(cid:105) = (cid:88) (cid:88) (cid:88) f∈Aik g∈Aj(cid:96) •∈f∩g ε•(f, g)f•g• We want to prove that this expression is zero. To do so, we will de(cid:27)ne a sign-reversing, (cid:27)xed-point-free permutation on the set of terms appearing in the sum. That is, each term which appears can be paired with another term with opposite coe(cid:28)cient and the same monomial in A(cid:92). We start by choosing an arbitrary non-zero term in the sum, of the form ε•(f, g) f•g•. To de(cid:27)ne a permutation as suggested above, we need to know what are the other terms in the sum that are equivalent to f•g• in A(cid:92). The answer is given by the following lemmas: Lemma 9. Suppose f•g• ≡ f(cid:48)∗g(cid:48)∗ mod [L•, L•], and that ε∗(f(cid:48), g(cid:48)) (cid:54)= 0. Then ∗ is in either f ∩ g, f ∩ f, or g ∩ g. Proof. Certainly f(cid:48) and g(cid:48) together have the same combined set of edges as f and g since f•g• = f(cid:48)∗g(cid:48)∗ cyclically. Then since two paths come together at ∗ (since ε∗(f(cid:48), g(cid:48)) (cid:54)= 0), it must be that either f and g meet at ∗, or f or g meets itself. Lemma 10. Let • and ∗ be as in the previous lemma, with f•g• = f(cid:48)∗g(cid:48)∗. Additionally assume that (f, g) and (f(cid:48), g(cid:48)) are both in Aik × Aj(cid:96). Then 90 (a) (b) If ∗ ∈ f ∩ g, then (f(cid:48), g(cid:48)) are obtained from (f, g) by swapping the segments of f and g between • and ∗. If ∗ ∈ f ∩ f, then (f(cid:48), g(cid:48)) are obtained from (f, g) by swapping the entire loop g with the subloop of f based at ∗. Proof. (a) Denote by a and b the segments of f between • and ∗, and similarly let x and y be the segments of g between • and ∗, so we may write f = •a ∗ b and g = •x ∗ y. Then f•g• = •a ∗ b • x ∗ y Since we assumed that f(cid:48)∗g(cid:48)∗ = f•g• up to cyclic permutation, then we can write it starting at ∗ as f(cid:48)∗g(cid:48)∗ = ∗b • x ∗ y • a This means f(cid:48) = ∗b • x and g(cid:48) = ∗y • a, or the other way around: f(cid:48) = ∗y • a and g(cid:48) = ∗b • x. The two possibilities di(cid:29)er by switching the roles of f(cid:48) and g(cid:48). But since we assume (f(cid:48), g(cid:48)) ∈ Aik × Aj(cid:96), and since Aik (cid:54)= Aj(cid:96), only one of the possibilities will be correct. After cyclically permuting, we have f(cid:48) = •x ∗ b and g(cid:48) = •a ∗ y (or the other way around). Thus we see that f(cid:48) and g(cid:48) are obtained from f and g by swapping either x with a or y with b, which are the portions of the paths in between their common subpaths • and ∗. This is illustrated in Figure 8.2. Note that since (f(cid:48), g(cid:48)) ∈ Aik × Aj(cid:96), the swapped segments must have the same intersection index with the cut. (b) Now suppose ∗ ∈ f ∩ f. Let a, b, c be the segments of f between • and ∗, so that f = 91 Figure 8.2: A “type I” swap g f • ∗ ←→ g(cid:48) f(cid:48) • ∗ •a ∗ b ∗ c. Also let x be the rest of g after •, so that g = •x. Then f•g• = •a ∗ b ∗ c • x Cyclically permuting, and using the assumption that f•g• = f(cid:48)∗g(cid:48)∗, we get f(cid:48)∗g(cid:48)∗ = ∗c • x • a ∗ b Again there are two possibilities. Either f(cid:48) = ∗c • x • a and g(cid:48) = ∗b, or f(cid:48) and g(cid:48) are (cid:30)ipped. But the condition that f(cid:48) ∈ Aik and g(cid:48) ∈ Aj(cid:96) ensures that only one of the two choices is correct. Suppose it is the (cid:27)rst case, and f(cid:48) = ∗c • x • a and g(cid:48) = ∗b. Then f(cid:48) is obtained from f by removing the loop based at ∗ and adding the loop g, and g(cid:48) is simply the subloop of f based at ∗. This is illustrated in Figure 8.3. Note that since (f(cid:48), g(cid:48)) ∈ Aik × Aj(cid:96), the loops •x and ∗b must cross the cut and rim the same number of times. g f • Figure 8.3: A “type II” swap ←→ g f • ∗ ∗ The next lemma is a converse to the previous one, so in fact the conditions given above com- 92 pletely characterize the set of terms in the expansion of(cid:10)tik, tj(cid:96) (cid:11) with a given term f•g•. The result is fairly clear, and we omit the proof. Lemma 11. Let (f, g) ∈ Aik × Aj(cid:96) and • ∈ f ∩ g such that ε•(f, g) (cid:54)= 0. (a) (b) If there exists an ∗ ∈ f ∩ g such that the segments of f and g between • and ∗ (or between ∗ and •) cross the cut and rim the same number of times, then swapping those segments gives (f(cid:48), g(cid:48)) ∈ Aik × Aj(cid:96) such that f(cid:48)∗g(cid:48)∗ = f•g•. If there exists an ∗ ∈ f ∩ f such that g and the subloop of f based at ∗ cross the cut and rim the same number of times, then swapping g and this subloop gives (f(cid:48), g(cid:48)) ∈ Aik × Aj(cid:96) such that f(cid:48)∗g(cid:48)∗ = f•g•. We will call the swaps from part (a) of the previous two lemmas “type I” swaps, and the swaps from part (b) will be called “type II” swaps. We are now ready to de(cid:27)ne our sign-reversing map, which we will denote σ, which acts on the set of terms appearing in the sum. Lemma 12. There exists a (cid:27)xed-point-free permutation σ of the non-zero terms appearing in the expansion of(cid:10)tik, tj(cid:96) (cid:11) such that σ(x) = −x for each term x. Proof. Recall our convention that f is a term in tik and g a term in tj(cid:96). Assume that i ≥ j, so that f goes around the torus more times “horizontally”. Starting at •, walk along f until we reach the (cid:27)rst admissible swap. If the (cid:27)rst admissible swap is of type I, and occurs at ∗, then we will de(cid:27)ne the image of the term ε•(f, g)f•g• under σ to be the term ε∗(f(cid:48), g(cid:48))f(cid:48)∗g(cid:48)∗, where f(cid:48) and g(cid:48) are obtained by performing the type I swap between • and ∗. In order for ε∗(f(cid:48), g(cid:48)) (cid:54)= 0, it must be that f and g cross at ∗ (rather than touch). This guarantees that ε•(f, g)f•g• is not a (cid:27)xed point of σ. To see that ε∗(f(cid:48), g(cid:48)) = −ε•(f, g), note that by the preceding lemmas, the segments of f and g from • to ∗ must cross the cut and rim the same number of times. This means that on the universal cover, the segments of f and g between • and ∗ bound a contractible disc. Thus after 93 performing the swap, f(cid:48) and g(cid:48) touch with opposite orientation – i.e. if f was to the left, and g to the right, then f(cid:48) is on the right, and g(cid:48) on the left. This can be seen in Figure 8.2. If, on the other hand, the (cid:27)rst admissible swap is of type II, we consider two separate cases. Recall that a type II swap means f intersects itself at ∗, so we may write f∗ = f(cid:48)∗f(cid:48)(cid:48)∗ , and that f(cid:48)(cid:48)∗ ∈ Aj(cid:96) (the same as g). First we consider the case that f(cid:48)(cid:48)∗ does not intersect g. In this case, we de(cid:27)ne the action of σ on ε•(f, g)f•g• to be the term corresponding to performing the type II swap. Now, consider the second case, in which f(cid:48)(cid:48)∗ does intersect g. In this case, let (cid:63) be the (cid:27)rst intersection point after ∗. Then we de(cid:27)ne the action of σ on ε•(f, g)f•g• to be the term corresponding to the type I swap from • to (cid:63). Theorem 16 now follows immediately as a corollary, since the lemma implies that there are an even number of terms equivalent to any given f•g•, half with coe(cid:28)cient +1 and half with coe(cid:28)cient −1. 94 Chapter 9 Recovering the Lax Representation In [FMB15], the authors gave a Lax representation of the Grassmann pentagram map, which gives a family of invariants. However, they did not formulate Liouville integrability, since there was no associated Poisson structure. In this section, we explain how to apply the results of the previous sections to recover the invariants from [FMB15], and additionally give a Poisson structure in which these invariants commute. As described earlier, we may lift the vertices of a twisted Grassmann n-gon to 3N-by-N matrices Vi, and we get a relation of the form Vi+3 = ViAi + Vi+1Bi + Vi+2Ci As before, we de(cid:27)ne the 3N-by-3N matrices Vi = (ViVi+1Vi+2). Then the formula above can be re-phrased by saying Vi+1 = ViLi, where Li is the block matrix  Li =  0 0 Ai IdN 0 Bi 0 IdN Ci Assuming the lift was twisted, then the monodromy matrix of the polygon is related to the Li by M Vi = Vi LiLi+1 ··· Li+n−1 95  Li(λ) = 0 IdN 0 0 Ai λ−1Bi 0 IdN λCi  (cid:101)Li(λ) = 0 IdN 0 0 λAi λBi 0 IdN Ci   In that paper, the authors proved the following. Theorem 17. [FMB15] The Grassmann pentagram map is invariant under the scaling Bi (cid:55)→ λ−1Bi, Ci (cid:55)→ λCi. In light of this theorem, we may de(cid:27)ne the modi(cid:27)ed matrix with parameter λ: In [FMB15], the authors proved that the conjugacy class of L(λ) := L1(λ)··· Ln(λ) is pre- served under the pentagram map, so the spectral invariants of L(λ) are invariants of the map. It is not hard to see that the pentagram map is also invariant under the scaling Ai (cid:55)→ λAi, Bi (cid:55)→ λBi. We will call the corresponding matrix(cid:101)Li(λ): In [GSTV16], the authors present a very similar Lax representation for the pentagram map on P2, and explicitly describe the connection with the boundary measurement matrix used in the combinatorial proof of integrability. We now mimic this approach to connect the earlier results from this paper to the Lax representation described above from [FMB15]. In order to more easily generalize this approach, we change some of our notations and conventions from earlier to more closely resemble those from [GSTV16]. In particular, we re-index the Yi’s by 1, so that our relation 96 reads Vi+3 = ViYi−1 + Vi+1Xi + Vi+2 Also, recall we use an oriented curve called the “cut” on the cylinder to determine the powers of λ in the boundary measurements. Earlier in the paper, we chose the convention that when we draw the cylinder as a rectangle (identifying the top/bottom edges), we take the cut to be the the top/bottom edge. We change this convention now so that the cut goes diagonally down and to the right, crossing each edge that connects two white vertices. With this convention, the quiver can be realized as the concatenation of the elementary networks pictured in Figure 9.1. In the (cid:27)gure, the cut is the dotted line. Figure 9.1: Elementary Networks 3 1 2 Xi Yi 2 3 1 The boundary measurement matrix of the ith elementary network is given by   Bi(λ) = 0 Xi Xi + Yi λ 0 0 1 0 1 The boundary measurement matrix of the entire network Qn is then the product B(λ) := B1(λ)··· Bn(λ). Although our matrices have entries in the non-commutative ring L•[λ±], the following result from [GSTV16] is true in this more general context: 97 Proposition 13. [GSTV16] Bi(λ) = Ai(λ)(cid:101)Li+1(λ)Ai+1(λ)−1, where the matrices Ai(λ) are given by Ai(λ) :=   λ−1 0 Xi 0 1 0 0 1 In particular, B(λ) is conjugate to(cid:101)L(λ) :=(cid:101)L1(λ)···(cid:101)Ln(λ). Therefore, the spectral invariants of B(λ) are the same as those of(cid:101)L(λ). The coe(cid:28)cients of 0 λ in the expansions of tr(B(λ)k) can be written as non-commutative polynomials in the Xi and Yi. Thus the (traces of the) spectral invariants can be interpreted as functions on GPn,N, after identifying it with GL2n+1 /Ad GLN, as in Theorem 6. The Poisson bracket is given by taking the trace of the induced bracket on L (cid:92)• : N {tr(Xi), tr(Yj)} = tr(cid:10)Xi, Yj (cid:11) (cid:80) As before, we let tij denote the components of the spectral invariants, so that tr(B(λ)j) = i tijλi. The discussion above tells us that tr(tij) may be interpreted as functions on GPn,N. Although in the present paper we chose a di(cid:29)erent normalization convention (namely the Xi, Yi, Z matrices) than Mari Be(cid:29)a and Felipe, these invariant functions are still essentially the same as those mentioned in [FMB15]. Theorem 16 then implies that these functions form an involutive family with respect to the induced Poisson structure. 98 Chapter 10 Conclusions and Further Questions We have shown in Theorem 15 and Theorem 16 that the coe(cid:28)cients tij of the traces of powers of the boundary measurement matrix are non-commutative invariants of the pentagram map, and that they form an involutive family under the induced Lie bracket (cid:104)−,−(cid:105) on L (cid:92). In this sense, we have established a form of non-commutative integrability for the Grassmann pentagram map, in terms of the Xi, Yi variables/matrices. Furthermore, this induces a Poisson structure on the moduli space GPn,N in which Mari-Be(cid:29)a and Felipe’s invariants Poisson-commute. However, in the usual commutative setting, the de(cid:27)nition of Liouville integrability requires not only a Poisson-commuting family of invariants, but also that these invariants are indepen- dent, and that they are a maximal family of such independent commuting invariants. These last two aspects — independence and maximality — were not addressed in the present paper. It would be nice to have a proof that among this in(cid:27)nite family of commuting invariants, one can (cid:27)nd a maximal independent subset, so that we have a Liouville integrable system. The proof of Theorem 16 is very combinatorial. It would be interesting to give an alternate proof using R-matrices, which would more closely resemble the method of proof in the classical case [GSTV16]. 99 REFERENCES 100 REFERENCES [Art18] [Ber08] [BG12] [CB11] [Coh77] [FG06] [FMB15] [FZ02] [FZ07] Semen Artamonov. Generalized quasi Poisson structures and noncommutative inte- grable systems. PhD thesis, Rutgers University-School of Graduate Studies, 2018. Michel Van Den Bergh. Double poisson algebras. Transactions of the American Math- ematical Society, 360(11):5711–5769, 2008. Ken Brown and Ken R Goodearl. Lectures on algebraic quantum groups. Birkhäuser, 2012. William Crawley-Boevey. Poisson structures on moduli spaces of representations. Journal of Algebra, 325(1):205–215, 2011. Paul Moritz Cohn. Skew (cid:27)eld constructions, volume 27. CUP Archive, 1977. Vladimir Fock and Alexander Goncharov. Moduli spaces of local systems and higher teichmüller theory. Publications Mathématiques de l’Institut des Hautes Études Scien- ti(cid:27)ques, 103(1):1–211, 2006. Raúl Felipe and Gloria Mari Be(cid:29)a. The pentagram map on grassmannians. arXiv preprint arXiv:1507.04765, 2015. Sergey Fomin and Andrei Zelevinsky. Cluster algebras i: foundations. Journal of the American Mathematical Society, 15(2):497–529, 2002. Sergey Fomin and Andrei Zelevinsky. Cluster algebras iv: coe(cid:28)cients. Compositio Mathematica, 143(1):112–164, 2007. [GGRW02] Israel Gelfand, Sergei Gelfand, Vladimir Retakh, and Robert Wilson. Quasidetermi- nants. arXiv preprint math/0208146, 2002. Kenneth Ralph Goodearl and Stephane Launois. The dixmier-moeglin equivalence and a gel’fand-kirillov problem for poisson polynomial algebras. Bulletin de la Société Mathématique de France, 139(1):1–39, 2011. Max Glick. 227(2):1019–1045, 2011. The pentagram map and y-patterns. Advances in Mathematics, [GL11] [Gli11] [Gol84] William M Goldman. The symplectic nature of fundamental groups of surfaces. Ad- vances in Mathematics, 54(2):200–225, 1984. 101 [Gol86] William M Goldman. surface group representations. Inventiones mathematicae, 85(2):263–302, 1986. Invariant functions on lie groups and hamiltonian (cid:30)ows of [GSTV16] Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, and Alek Vainshtein. In- tegrable cluster dynamics of directed networks and pentagram maps. Advances in Mathematics, 300:390–450, 2016. [GSV09] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Poisson geometry of di- rected networks in a disk. Selecta Mathematica, New Series, 15(1):61–103, 2009. [GSV10a] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Cluster algebras and Pois- son geometry. Number 167. American Mathematical Soc., 2010. [GSV10b] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Poisson geometry of di- [Izo18] [Lew74] rected networks in an annulus. arXiv preprint arXiv:0901.0020, 2010. Anton Izosimov. Pentagram maps and refactorization in poisson-lie groups. arXiv preprint arXiv:1803.00726, 2018. Jacques Lewin. Fields of fractions for group algebras of free groups. Transactions of the American Mathematical Society, 192:339–346, 1974. [LGPV12] Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke. Poisson structures, volume 347. Springer Science & Business Media, 2012. [MKS76] Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinatorial group theory. Dover Publications, 1976. [MO17] [MT12] [OST10] [Pen12] [Pos06] John Machacek and Nicholas Ovenhouse. Log-canonical coordinates for poisson brackets and rational changes of coordinates. Journal of Geometry and Physics, 121:288–296, 2017. Gwenael Massuyeau and Vladimir Turaev. Quasi-poisson structures on represen- tation spaces of surfaces. International Mathematics Research Notices, 2014(1):1–64, 2012. Valentin Ovsienko, Richard Schwartz, and Serge Tabachnikov. The pentagram map: a discrete integrable system. Communications in Mathematical Physics, 299(2):409– 446, 2010. Robert C Penner. Decorated Teichmüller theory, volume 1. European Mathematical Society, 2012. Alexander Postnikov. Total positivity, grassmannians, and networks. arXiv preprint math/0609764, 2006. 102 [Pro76] [Sch92] [Sib68] Claudio Procesi. The invariant theory of n × n matrices. Advances in Mathematics, 19(3):306 – 381, 1976. Richard Schwartz. The pentagram map. Experimental Mathematics, 1(1):71–81, 1992. KS Sibirskii. Algebraic invariants for a set of matrices. Siberian Mathematical Journal, 9(1):115–124, 1968. 103