Cluster Algebras and Compatible Poisson Structures

Cluster Algebras and Compatible Poisson Structures Poisson 2012, Utrecht July, 2012 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 1 / 96

(Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 2 / 96

Reference: Cluster algebras and Poisson Geometry, M.Gekhtman, M.Shapiro, A.Vainshtein, AMS Surveys and Monographs, 2010 and references therein http://www.math.lsa.umich.edu/~fomin/cluster.html (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 2 / 96

Motivation: Totally Positive Matrices Totally Positive Matrices An n n matrix A is totally positive if all its minors are positive. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 3 / 96

Motivation: Totally Positive Matrices Totally Positive Matrices An n n matrix A is totally positive if all its minors are positive. Note that the number of all minors grows exponentially with size. However, one can select (not uniquely) a family F of just n 2 minors of A such that A is totally positive iff every minor in the family is positive. (Berenshtein-Fomin-Zelevinsky) (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 3 / 96

n = 3 Motivation: Totally Positive Matrices For n = 3 (total # of minors is 20), Families F 1 and F 2 F 1 = { 3 3, 23 23, 13 23, 23 13; 3 1, 1 3, 23 12, 12 23, 123 123} F 2 = { 3 3, 13 23, 23 13, 13 13; 3 1, 1 3, 23 12, 12 23, 123 123} (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 4 / 96

n = 3 Motivation: Totally Positive Matrices For n = 3 (total # of minors is 20), F 1 = { 3 3, 23 23, 13 23, 23 13; 3 1, 1 3, 23 12, 12 23, 123 123} F 2 = { 3 3, 13 23, 23 13, 13 13; 3 1, 1 3, 23 12, 12 23, 123 123} Families F 1 and F 2 differ in only one element (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 4 / 96

n = 3 Motivation: Totally Positive Matrices For n = 3 (total # of minors is 20), F 1 = { 3 3, 23 23, 13 23, 23 13; 3 1, 1 3, 23 12, 12 23, 123 123} F 2 = { 3 3, 13 23, 23 13, 13 13; 3 1, 1 3, 23 12, 12 23, 123 123} Families F 1 and F 2 differ in only one element are connected by 13 13 23 23 = 23 13 13 23 + 3 3 123 123 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 4 / 96

Motivation: Totally Positive Matrices Properties Every other test family of 9 minors (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 5 / 96

Motivation: Totally Positive Matrices Properties Every other test family of 9 minors contains 3 1, 1 3, 23 12, 12 23, 123 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 5 / 96

Motivation: Totally Positive Matrices Properties Every other test family of 9 minors contains 3 1, 1 3, 23 12, 12 23, 123 can be obtained from F 1 (or F 2 ) via a sequence of similar transformations defines coordinate system on GL(3) bi-rationally related to natural coordinates A = (a ij ) 3 i,j=1. The intersection of opposite big Bruhat cells coincides with B + w 0 B + B w 0 B GL(3) {A GL(3) 3 1 1 3 23 12 12 23 123 123 0} The number of connected components of this intersection can be computed using families F i. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 5 / 96

Instructive example Homogeneous coordinate ring C[Gr 2,n+3 ] Gr 2,n+3 ={V C n+3 :dim(v )=2}. The ring A = C[Gr 2,n+3 ] is generated by the Plücker coordinates x ij, for 1 i < j n + 3. Relations: x ik x jl =x ij x kl +x il x jk, for i <j <k <l. 3 x 23 x 34 2 x 24 4 x 13 x 35 x 12 x 25 x 14 x 45 1 x 15 5 sides: scalars diagonals: cluster variables relations: flips clusters: triangulations Each cluster has exactly n elements, so A is a cluster algebra of rank n. The monomials involving non-crossing variables form a linear basis in A (studied in [Kung-Rota]). (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 6 / 96

Instructive example Double Bruhat cell in SL(3) A = C[G u,v ], where G u,v = BuB B vb = x α 0 = γ y β SL 3 (C) : 0 δ z is a double Bruhat cell (u,v S 3, l(u)=l(v)=2). Ground ring: A = C[α ±1, β ±1, γ ±1, δ ±1 ]. Five cluster variables. Exchange relations: xy = x α + αγ yz = y γ y δ α 0 β 0 γ 0 δ 0 β + βδ z x y δ β = αγz + 1 z x α y γ y δ β z z x α = βδx + 1 γ y = αβγδ + y. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 7 / 96

Definition of Cluster Algebras (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 8 / 96

Definition of Cluster Algebras Definition Exchange graph of a cluster algebra: vertices clusters edges exchanges. T m m-regular tree with {1, 2,... m}-labeled edges, adjacent edges receive different labels (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 8 / 96

Definition of Cluster Algebras Definition Exchange graph of a cluster algebra: vertices clusters edges exchanges. T m m-regular tree with {1, 2,... m}-labeled edges, adjacent edges receive different labels 1 T 1 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 8 / 96

Definition of Cluster Algebras Definition Exchange graph of a cluster algebra: vertices clusters edges exchanges. T m m-regular tree with {1, 2,... m}-labeled edges, adjacent edges receive different labels 1 T 1 1 2 1 2 1 2 T 2 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 8 / 96

Definition of Cluster Algebras T 3 3 3 2 1 2 1 2 1 3 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 9 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) frozen variables, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) x(t) = (x 1 (t),..., x m (t)) frozen variables, cluster variables, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) frozen variables, x(t) = (x 1 (t),..., x m (t)) cluster variables, z(t) = (z 1 (t),..., z n (t)) = (x(t), y) extended cluster variables. Definition Cluster algebra A is given by pair (B(t), z(t)) for each cluster (vertex of exchange graph) t B(t) is an m n integral matrix (m n) whose left m m block is left-skew-symmetrizable, (we will assume it skew-symmetric for simplicity) (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) frozen variables, x(t) = (x 1 (t),..., x m (t)) cluster variables, z(t) = (z 1 (t),..., z n (t)) = (x(t), y) extended cluster variables. Definition Cluster algebra A is given by pair (B(t), z(t)) for each cluster (vertex of exchange graph) t B(t) is an m n integral matrix (m n) whose left m m block is left-skew-symmetrizable, (we will assume it skew-symmetric for simplicity) z(t) is a vector of extended cluster variables. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) frozen variables, x(t) = (x 1 (t),..., x m (t)) cluster variables, z(t) = (z 1 (t),..., z n (t)) = (x(t), y) extended cluster variables. Definition Cluster algebra A is given by pair (B(t), z(t)) for each cluster (vertex of exchange graph) t B(t) is an m n integral matrix (m n) whose left m m block is left-skew-symmetrizable, (we will assume it skew-symmetric for simplicity) z(t) is a vector of extended cluster variables. variables z m+1 = y 1,..., z n = y n m are not affected by T i. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster algebras of geometric type y = (y 1,..., y n m ) frozen variables, x(t) = (x 1 (t),..., x m (t)) cluster variables, z(t) = (z 1 (t),..., z n (t)) = (x(t), y) extended cluster variables. Definition Cluster algebra A is given by pair (B(t), z(t)) for each cluster (vertex of exchange graph) t B(t) is an m n integral matrix (m n) whose left m m block is left-skew-symmetrizable, (we will assume it skew-symmetric for simplicity) z(t) is a vector of extended cluster variables. variables z m+1 = y 1,..., z n = y n m are not affected by T i. both B(t) and z(t) are subject to cluster transformations defined as follows. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 10 / 96

Definition of Cluster Algebras Cluster transformations (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 11 / 96

Definition of Cluster Algebras Cluster transformations Cluster change t i t For an edge of T m i [1,..., m] T i : z(t) z(t ) is defined as x i (t ) = 1 x i (t) b ik (t)>0 z k (t) b ik (t) + z j (t ) = z j (t) j i, b ik (t)<0 z k (t) b ik (t) (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 11 / 96

Definition of Cluster Algebras Cluster transformations Cluster change t i t For an edge of T m i [1,..., m] T i : z(t) z(t ) is defined as x i (t ) = 1 x i (t) b ik (t)>0 z k (t) b ik (t) + z j (t ) = z j (t) j i, b ik (t)<0 z k (t) b ik (t) Matrix mutation B(t ) = T i (B(t)), b kl (t), if (k i)(l i) = 0 b kl (t ) = b kl (t) + b ki(t) b il (t) + b ki (t) b il (t), otherwise. 2 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 11 / 96

Definition of Cluster Algebras (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 12 / 96

Definition of Cluster Algebras Definition Given some initial cluster t 0 put z i = z i (t 0 ), B = B(t 0 ). The cluster algebra A (or, A(B)) is the subalgebra of the field of rational functions in cluster variables z 1,..., z n generated by the union of all cluster variables z i (t). (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 12 / 96

Definition of Cluster Algebras Examples of T i (B) A matrix B(t) can be represented by a (weighted, oriented) graph. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 13 / 96

Definition of Cluster Algebras The Laurent phenomenon Theorem (FZ) In a cluster algebra, any cluster variable is expressed in terms of initial cluster as a Laurent polynomial. Positivity Conjecture All these Laurent polynomials have positive integer coefficients (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 14 / 96

Definition of Cluster Algebras Examples of Cluster Transformations (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 15 / 96

Definition of Cluster Algebras Examples of Cluster Transformations Short Plücker relation in G k (n) x ijj x klj = x ikj x jlj + x ilj x kjj for 1 i < k < j < l m, J = k 2. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 15 / 96

Definition of Cluster Algebras Examples of Cluster Transformations Short Plücker relation in G k (n) x ijj x klj = x ikj x jlj + x ilj x kjj for 1 i < k < j < l m, J = k 2. Whitehead moves and Ptolemy relations in Decorated Teichmüller space: P b Q P b Q a p c a q c d d M N M N f (p)f (q) = f (a)f (c) + f (b)f (d) (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 15 / 96

Compatible Poisson structure and 2-form τ-coordinates Nondegenerate coordinate change: { j i τ i (t) = z j(t) b ij (t) for i m, j i z j(t) b ij (t) /z i (t) for m + 1 i n. Exchange in direction i: τ i 1 τ i ; τ j (1 + τ i b ij, if b ij > 0, τ j ( ) bij τ τi j 1+τ i, otherwise. Definition We say that a skew-symmetrizable matrix A is reducible if there exists a permutation matrix P such that PAP T is a block-diagonal matrix, and irreducible otherwise. The reducibility ρ(a) is defined as the maximal number of diagonal blocks in PAP T. The partition into blocks defines an obvious equivalence relation on the rows (or columns) of A. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 16 / 96

Compatible Poisson structure and 2-form Compatible Poisson structures A Poisson bracket {, } is compatible with the cluster algebra A if, for any extended cluster z = (z 1,..., z n ) {z i, z j } = ω ij z i z j, where ω ij Z are constants for all i, j [1, n + m]. Theorem For an B Z n,n+m as above of rank n the set of compatible Poisson brackets has dimension ρ(b) + ( ) m 2. Moreover, the coefficient matrices Ω τ of these Poisson brackets in the basis τ are characterized by the equation Ω τ [m, n] = ΛB for some diagonal matrix Λ = diagonal(λ 1,..., λ n ) where λ i = λ j whenever i j. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 17 / 96

Compatible Poisson structure and 2-form Degenerate exchange matrix Example Cluster algebra of rank 3 with trivial coefficients. Exchange matrix B = 0 1 1 1 0 1. Compatible Poisson bracket must satisfy 1 1 0 {x 1, x 2 } = λx 1 x 2, {x 1, x 3 } = µx 1 x 3, {x 2, x 3 } = νx 2 x 3 Exercise: Check that these conditions imply λ = µ = ν = 0. Conclusion: Only trivial Poisson structure is compatible with the cluster algebra. What to do? We will use the dual language of 2-forms (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 18 / 96

Compatible Poisson structure and 2-form Compatible 2-forms Definition 2-form ω is compatible with a collecion of functions {f i } if ω = i,j ω ij df i f i df j f j Definition 2-form ω is compatible with a cluster algebra if it is compatible with all clusters. Exercise Check that the form ω = dx 1 with the example above. x 1 dx 2 x 2 dx 1 x 1 dx 3 x 3 + dx 2 x 2 dx 3 x 3 is compatible (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 19 / 96

Compatible Poisson structure and 2-form Compatible 2-forms Theorem For an B Z n,n+m the set of Poisson brackets for which all extended clusters in A(B) are log-canonical has dimension ρ(b) + ( m 2). Moreover, the coefficient matrices Ω x of these 2-forms in initial cluster are characterized by the equation Ω x [m, n] = ΛB, where Λ = diagonal(λ 1,..., λ n ), with λ i = λ j 0 whenever i j. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 20 / 96

Compatible Poisson structure and 2-form Cluster manifold For an abstract cluster algebra of geometric type A of rank m we construct an algebraic variety A (which we call cluster manifold) Idea: A is a good part of Spec(A). We will describe A by means of charts and transition functions. For each cluster t we define an open chart A(t) = Spec(C[x(t), x(t) 1, y]), where x(t) 1 means x 1 (t) 1,..., x m (t) 1. Transitions between charts are defined by exchange relations x i (t )x i (t) = z k (t) b ik (t) + z k (t) b ik (t) b ik (t)>0 z j (t ) = z j (t) j i, b ik (t)<0 Finally, A = t A(t). (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 21 / 96

Compatible Poisson structure and 2-form Nonsingularity of A A contains only such points p Spec(A) that there is a cluster t whose cluster elements form a coordinate system in some neighborhood of p. Observation The cluster manifold A is nonsingular and possesses a Poisson bracket that is log-canonical w.r.t. any extended cluster. Let ω be one of these Poisson brackets. Casimir of ω is a function that is in involution with all the other functions on A. All rational casimirs form a subfield F C in the field of rational functions C(A). The following proposition provides a complete description of F C. Lemma F C = F (m 1,..., m s ), where m j = y α ji i for some integral α ji, and s = corankω. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 22 / 96

Compatible Poisson structure and 2-form Toric action We define a local toric action on the extended cluster t as the C -action given by the formula z i (t) z i (t) ξ w i (t), ξ C for some integral w i (t) (called weights of toric action). Local toric actions are compatible if taken in all clusters they define a global action on A. This toric action is said to be an extension of the above local actions. A 0 is the regular locus for all compatible toric actions on A. A 0 is given by inequalities y i 0. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 23 / 96

Compatible Poisson structure and 2-form Symplectic leaves A is foliated into a disjoint union of symplectic leaves of ω. Given generators q 1,..., q s of the field of rational casimirs F C we have a map Q : A C s, Q(x) = (q 1 (x),..., q s (x)). We say that a symplectic leaf L is generic if there exist s vector fields u i on A such that a) at every point x L, the vector u i (x) is transversal to the surface Q 1 (Q(L)); b) the translation along u i for a sufficiently small time t gives a diffeomorphism between L and a close symplectic leaf L t. Lemma A 0 is foliated into a disjoint union of generic symplectic leaves of the Poisson bracket ω. Remark Generally speaking, A 0 does not coincide with the union of all generic symplectic leaves in A. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 24 / 96

Compatible Poisson structure and 2-form Connected components of A 0 Question: find the number #(A 0 ) of connected components of A 0. Let F n 2 be an n-dimensional vector space over F 2 with a fixed basis {e i }. Let B be a n n- matrix with Z 2 entries defined by the relation B B(t) (mod 2) for some cluster t, and let ω = ω t be a (skew-)symmetric bilinear form on F n 2, such that ω(e i, e j ) = b ij. Define a linear operator t i : F n 2 F n 2 by the formula t i(θ) = ξ ω(θ, e i )e i, and let Γ = Γ t be the group generated by t i, 1 i m. Theorem The number of connected components #(A 0 ) equals to the number of Γ t -orbits in F n 2. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 25 / 96

A and X manifolds Cluster A and X manifolds Coordinate ring A cluster algebra with cluster coordinates x i π X algebra generated by τ i Manifold A cluster variety with compatible 2-form π:τ i = n+m j=1 xb ij j X Poisson cluster variety (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 26 / 96

A and X manifolds Let Ω be a 2-form. KerΩ = {vector ξ : Ω(ξ, η) = 0 η} provides a fibration of the underlying vector space. {space of fibers of KerΩ} Im π is a local diffeomorphism. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 27 / 96

A and X manifolds Let Ω be a 2-form. KerΩ = {vector ξ : Ω(ξ, η) = 0 η} provides a fibration of the underlying vector space. {space of fibers of KerΩ} Im π is a local diffeomorphism. More generally, let Ω be a compatible 2-form on a cluster manifold A of coefficient-free cluster algebra A. Ker Ω determines an integrable distribution in T A. Generic fibers of KerΩ form a smooth manifold X whose dimension is rank(b). π : A X is a natural projection. Then, Ω = π (Ω) is a symplectic form on X dual to the Poisson structure. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 27 / 96

Applications and Examples Application of Compatible Poisson Structures If a Poisson variety (M, {, }) possesses a coordinate chart that consists of regular functions whose logarithms have pairwise constant Poisson brackets, then one can use this chart to define a cluster algebra A M that is closely related (and, under rather mild conditions, isomorphic) to the ring of regular functions on M and such that {, } is compatible with A M. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 28 / 96

Examples Applications and Examples (Decorated) Teichmüller space has a natural structure of cluster algebra. Weyl-Petersson symplectic form is the unique symplectic form compatible with the structure of cluster algebra. There exists a cluster algebra structure on SL n compatible with Sklyanin Poisson bracket. A 0 is the maximal double Bruhat cell. There exists a cluster algebra structure on Grassmanian compatible with push-forward of Sklyanin Poisson bracket. A 0 determined by the inequalities {solid Plücker coordinate 0}. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 29 / 96

Examples Applications and Examples (Decorated) Teichmüller space has a natural structure of cluster algebra. Weyl-Petersson symplectic form is the unique symplectic form compatible with the structure of cluster algebra. There exists a cluster algebra structure on SL n compatible with Sklyanin Poisson bracket. A 0 is the maximal double Bruhat cell. There exists a cluster algebra structure on Grassmanian compatible with push-forward of Sklyanin Poisson bracket. A 0 determined by the inequalities {solid Plücker coordinate 0}. We will now provide a detailed discussion of the last two examples. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 29 / 96

Poisson-Lie Groups Poisson-Lie Groups Let G be a Lie group. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 30 / 96

Poisson-Lie Groups Poisson-Lie Groups Let G be a Lie group. Definition The Poisson structure {, } on G is called Poisson-Lie if the multiplication map m : G G G is Poisson. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 30 / 96

Poisson-Lie Groups Poisson-Lie Groups Let G be a Lie group. Definition The Poisson structure {, } on G is called Poisson-Lie if the multiplication map m : G G G is Poisson. Example Sl 2. Borel subgroup B Sl 2 is the set Poisson structure on B: {t, x} = tx. {( t x )} 0 t 1 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 30 / 96

Poisson-Lie Groups Poisson-Lie Groups Let G be a Lie group. Definition The Poisson structure {, } on G is called Poisson-Lie if the multiplication map m : G G G is Poisson. Example {( )} t x Sl 2. Borel subgroup B Sl 2 is the set 0 t 1 Poisson structure on B: {t, x} = tx. {( ) ( )} t1 x Induced Poisson structure on B B = 1 t2 x 0 t1 1, 2 0 t2 1 : {t 1, x 1 } = t 1 x 1, {t 2, x 2 } = t 2 x 2. All other brackets are 0. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 30 / 96

Poisson-Lie Groups Poisson-Lie Groups Let G be a Lie group. Definition The Poisson structure {, } on G is called Poisson-Lie if the multiplication map m : G G G is Poisson. Example {( )} t x Sl 2. Borel subgroup B Sl 2 is the set 0 t 1 Poisson structure on B: {t, x} = tx. {( ) ( )} t1 x Induced Poisson structure on B B = 1 t2 x 0 t1 1, 2 0 t2 1 : {t 1, x 1 } = t 1 x 1, {t 2, x 2 } = t 2 x 2. All other brackets are 0. ( ) ( ) t1 x 1 t2 x 0 t1 1 2 0 t2 1 = ( t1 t 2 t 1 x 2 + x 1 t 1 2 0 t1 1 t 1 2 ) = ( u v ) 0 u 1 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 30 / 96

Poisson-Lie Groups For coordinates u, v (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 31 / 96

Poisson-Lie Groups For coordinates u, v {m (u), m (v)} G G = {t 1 t 2, t 1 x 2 + x 1 t 1 2 } G G = t 2 1 t 2x 2 + t 1 x 1. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 31 / 96

Poisson-Lie Groups For coordinates u, v {m (u), m (v)} G G = {t 1 t 2, t 1 x 2 + x 1 t2 1 } G G = t1 2t 2x 2 + t 1 x 1. On the other hand, m ({u, v} G ) = m (uv) = t 2 1t 2 x 2 + t 1 x 1, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 31 / 96

Poisson-Lie Groups For coordinates u, v {m (u), m (v)} G G = {t 1 t 2, t 1 x 2 + x 1 t2 1 } G G = t1 2t 2x 2 + t 1 x 1. On the other hand, m ({u, v} G ) = m (uv) = t 2 1t 2 x 2 + t 1 x 1, which proves Poisson-Lie property. Similarly, we define Poisson-Lie bracket for B. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 31 / 96

Poisson-Lie Groups For coordinates u, v {m (u), m (v)} G G = {t 1 t 2, t 1 x 2 + x 1 t2 1 } G G = t1 2t 2x 2 + t 1 x 1. On the other hand, m ({u, v} G ) = m (uv) = t 2 1t 2 x 2 + t 1 x 1, which proves Poisson-Lie property. Similarly, we define Poisson-Lie bracket for B. Then, if we have embedded Poisson subgroups B and B they define a Poisson-Lie structure on SL 2 they generate. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 31 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition of the open dense part of SL 2 into B B +. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition ( of ) the ( open dense ) part ( of SL 2 into B B + ). t1 0 t2 x Indeed, 2 t1 t y 1 t1 1 0 t2 1 = 2 t 1 x 2 y 1 t 2 y 1 x 2 + t1 1 t 1 2 Hence, {z 11, z 12 } = {t 1 t 2, t 1 x 2 } = t 2 1t 2 x 2 = z 11 z 12, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition ( of ) the ( open dense ) part ( of SL 2 into B B + ). t1 0 t2 x Indeed, 2 t1 t y 1 t1 1 0 t2 1 = 2 t 1 x 2 y 1 t 2 y 1 x 2 + t1 1 t 1 2 Hence, {z 11, z 12 } = {t 1 t 2, t 1 x 2 } = t 2 1t 2 x 2 = z 11 z 12, {z 11, z 21 } = {t 1 t 2, y 1 t 2 } = t 2 2t 1 y 1 = z 11 z 21, {z 11, z 22 } = {t 1 t 2, y 1 x 2 + t 1 1 t 1 2 } = 2t 1y 1 t 2 x 2 = 2z 12 z 21, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition ( of ) the ( open dense ) part ( of SL 2 into B B + ). t1 0 t2 x Indeed, 2 t1 t y 1 t1 1 0 t2 1 = 2 t 1 x 2 y 1 t 2 y 1 x 2 + t1 1 t 1 2 Hence, {z 11, z 12 } = {t 1 t 2, t 1 x 2 } = t 2 1t 2 x 2 = z 11 z 12, {z 11, z 21 } = {t 1 t 2, y 1 t 2 } = t 2 2t 1 y 1 = z 11 z 21, {z 11, z 22 } = {t 1 t 2, y 1 x 2 + t1 1 2 } = 2t 1y 1 t 2 x 2 = 2z 12 z 21, {z 12, z 21 } = {t 1 x 2, y 1 t 2 } = 0, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition ( of ) the ( open dense ) part ( of SL 2 into B B + ). t1 0 t2 x Indeed, 2 t1 t y 1 t1 1 0 t2 1 = 2 t 1 x 2 y 1 t 2 y 1 x 2 + t1 1 t 1 2 Hence, {z 11, z 12 } = {t 1 t 2, t 1 x 2 } = t 2 1t 2 x 2 = z 11 z 12, {z 11, z 21 } = {t 1 t 2, y 1 t 2 } = t 2 2t 1 y 1 = z 11 z 21, {z 11, z 22 } = {t 1 t 2, y 1 x 2 + t1 1 2 } = 2t 1y 1 t 2 x 2 = 2z 12 z 21, {z 12, z 21 } = {t 1 x 2, y 1 t 2 } = 0, {z 12, z 22 } = {t 1 x 2, y 1 x 2 + t 1 1 t 1 2 } = t 1y 1 t 2 x 2 2 + x 2 /t 2 = z 12 z 22, (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups To define Poisson-Lie bracket on the whole SL 2 we use Gauss decomposition ( of ) the ( open dense ) part ( of SL 2 into B B + ). t1 0 t2 x Indeed, 2 t1 t y 1 t1 1 0 t2 1 = 2 t 1 x 2 y 1 t 2 y 1 x 2 + t1 1 t 1 2 Hence, {z 11, z 12 } = {t 1 t 2, t 1 x 2 } = t 2 1t 2 x 2 = z 11 z 12, {z 11, z 21 } = {t 1 t 2, y 1 t 2 } = t 2 2t 1 y 1 = z 11 z 21, {z 11, z 22 } = {t 1 t 2, y 1 x 2 + t1 1 2 } = 2t 1y 1 t 2 x 2 = 2z 12 z 21, {z 12, z 21 } = {t 1 x 2, y 1 t 2 } = 0, {z 12, z 22 } = {t 1 x 2, y 1 x 2 + t1 1 t 1 2 } = t 1y 1 t 2 x2 2 + x 2 /t 2 = z 12 z 22, {z 21, z 22 } = {y 1 t 2, y 1 x 2 + t1 1 t 1 2 } = y 1 2 t 2 x 2 + y 1 /t 1 = z 21 z 22. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 32 / 96

Poisson-Lie Groups Poisson-Lie bracket for SL n Standard embeddings SL 2 SL n define Poisson submanifold with respect to standard Poisson-Lie bracket. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 33 / 96

Poisson-Lie Groups Poisson-Lie bracket for SL n Standard embeddings SL 2 SL n define Poisson submanifold with respect to standard Poisson-Lie bracket. Any fixed reduced decomposition of the maximal element of the Weyl group determines a Poisson map ( n 1 2 ) 1 SL 2 SL n.then, for X = (x ij ) SL n we have {x ij, x ki } = x ij x ik for j < k, {x ji, x ki } = x ji x ki for j < k {x ij, x kl } = x il x kj for i < k, j < l, {x ij, x kl } = 0 for i < k, j > l (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 33 / 96

R-matrix Poisson-Lie Groups One can construct a Poisson-Lie bracket using R matrix. Definition A map R : g g is called a classical R matrix if it satisfies modified Yang-Baxter equation [R(ξ), R(η)] R ([R(ξ), η] + [ξ, R(η)]) = [ξ, η] (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 34 / 96

R-matrix Poisson-Lie Groups One can construct a Poisson-Lie bracket using R matrix. Definition A map R : g g is called a classical R matrix if it satisfies modified Yang-Baxter equation [R(ξ), R(η)] R ([R(ξ), η] + [ξ, R(η)]) = [ξ, η] R-matrix Poisson bracket R-matrix Poisson-Lie bracket on SL n : {f 1, f 2 }(X ) = 1 2 ( R( f 1(X )X ), f 2 (X )X R(X f 1 (X )), X f 2 (X ))] ), where gradient f sl n defined w.r.t. trace form. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 34 / 96

Poisson-Lie Groups Example For any matrix X we write its decomposition into a sum of lower triangular and strictly upper triangular matrices as X = X + X 0 + X + The standard R-matrix R : Mat n Mat n defined by R(X ) = X + X The standard R-matrix Poisson-Lie bracket: {x ij, x αβ }(X ) = 1 2 (sign(α i) + sign(β j))x iβx αj (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 35 / 96

Poisson-Lie Groups Poisson Homogeneous Spaces X is a homogeneous space of an algebraic group G, i.e., m : G X X. G is equipped with Poisson-Lie structure. Definition Poisson bracket on X equips X with a structure of a Poisson homogeneous space if m is a Poisson map. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 36 / 96

Example: Cluster Structure on Grassmannians Grassmannian G k (n) Grassmannian G k (n) of k-dimensional subspaces of n-dimensional space. SL n acts freely on G k (n). (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 37 / 96

Example: Cluster Structure on Grassmannians Grassmannian G k (n) Grassmannian G k (n) of k-dimensional subspaces of n-dimensional space. SL n acts freely on G k (n). Maximal Schubert cell G 0 k (n) G k(n) contains elements of the form ( 1 Y ) where Y = (yij ), i [1, k]; j [1, n k]. Poisson homogeneous bracket w.r.t. the standard Poisson-Lie bracket on SL n : {y ij, y α,β } = 1 2 ((sign(α i) sign(β j)) y iβy α,j (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 37 / 96

Example: Cluster Structure on Grassmannians Grassmannian G k (n) Grassmannian G k (n) of k-dimensional subspaces of n-dimensional space. SL n acts freely on G k (n). Maximal Schubert cell G 0 k (n) G k(n) contains elements of the form ( 1 Y ) where Y = (yij ), i [1, k]; j [1, n k]. Poisson homogeneous bracket w.r.t. the standard Poisson-Lie bracket on SL n : {y ij, y α,β } = 1 2 ((sign(α i) sign(β j)) y iβy α,j To find a cluster structure in G k (n), need to find a coordinate system compatible with the bracket above. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 37 / 96

Example: Cluster Structure on Grassmannians Cluster algebra structure on Grassmannians (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 38 / 96

Example: Cluster Structure on Grassmannians Cluster algebra structure on Grassmannians The open cell in G k (n) : (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 38 / 96

Example: Cluster Structure on Grassmannians Cluster algebra structure on Grassmannians The open cell in G k (n) : G 0 k (n) = {X G k(n) : X = [1 k Y ]} (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 38 / 96

Example: Cluster Structure on Grassmannians Cluster algebra structure on Grassmannians The open cell in G k (n) : G 0 k (n) = {X G k(n) : X = [1 k Y ]} Initial extended cluster : F ij = ( 1) (k i)(l(i,j) 1) Y [j,j+l(i,j)] [i l(i,j),i], l(i, j) = min(i 1, n k j) (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 38 / 96

Example: Cluster Structure on Grassmannians Cluster algebra structure on Grassmannians The open cell in G k (n) : G 0 k (n) = {X G k(n) : X = [1 k Y ]} Initial extended cluster : F ij = ( 1) (k i)(l(i,j) 1) Y [j,j+l(i,j)] [i l(i,j),i], l(i, j) = min(i 1, n k j) j j Y F ij i F i j i (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 38 / 96

Example: Cluster Structure on Grassmannians Initial exchange matrix B (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 39 / 96

Example: Cluster Structure on Grassmannians Initial exchange matrix B : a (0, ±1)-matrix represented by a directed graph (quiver) 1 2 3 4 m 4 m 3 m 2 m 1 m (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 39 / 96

Example: Cluster Structure on Grassmannians Initial exchange matrix B : a (0, ±1)-matrix represented by a directed graph (quiver) 1 2 3 4 m 4 m 3 m 2 m 1 m Initial cluster transformations (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 39 / 96

Example: Cluster Structure on Grassmannians Initial exchange matrix B : a (0, ±1)-matrix represented by a directed graph (quiver) 1 2 3 4 m 4 m 3 m 2 m 1 m Initial cluster transformations are built out of short Plücker relations. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 39 / 96

And now for something completely different... (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 40 / 96

Planar Networks (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 41 / 96

Planar Networks Weighted network N in a disk (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 41 / 96

Planar Networks Weighted network N in a disk G = (V, E) - directed planar graph drawn inside a disk with the vertex set V and the edge set E. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 41 / 96

Planar Networks Weighted network N in a disk G = (V, E) - directed planar graph drawn inside a disk with the vertex set V and the edge set E. Exactly n of its vertices are located on the boundary circle of the disk. They are labelled counterclockwise b 1,..., b n and called boundary vertices. Each boundary vertex is labelled as a source or a sink. I = {i 1,..., i k } [1, n] is a set of sources. J = [1, n] \ I - set of sinks. All the internal vertices of G have degree 3 and are of two types: either they have exactly one incoming edge, or exactly one outcoming edge. The vertices of the first type are called white, those of the second type, black. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 41 / 96

Planar Networks Weighted network N in a disk G = (V, E) - directed planar graph drawn inside a disk with the vertex set V and the edge set E. Exactly n of its vertices are located on the boundary circle of the disk. They are labelled counterclockwise b 1,..., b n and called boundary vertices. Each boundary vertex is labelled as a source or a sink. I = {i 1,..., i k } [1, n] is a set of sources. J = [1, n] \ I - set of sinks. All the internal vertices of G have degree 3 and are of two types: either they have exactly one incoming edge, or exactly one outcoming edge. The vertices of the first type are called white, those of the second type, black. To each e E we assign a weight w e. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 41 / 96

Example Planar Networks b 1 b 4 w w w w 1 2 3 4 w w w 9 10 11 l w w 5 6 7 8 w w b 2 b 3 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 42 / 96

Planar Networks Boundary Measurements (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Planar Networks Boundary Measurements Paths and cycles A path P in N is an alternating sequence (v 1, e 1, v 2,..., e r, v r+1 ) of vertices and edges such that e i = (v i, v i+1 ) for any i [1, r]. A path is called a cycle if v r+1 = v 1 Concordance number rotation number (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Planar Networks Boundary Measurements Paths and cycles A path P in N is an alternating sequence (v 1, e 1, v 2,..., e r, v r+1 ) of vertices and edges such that e i = (v i, v i+1 ) for any i [1, r]. A path is called a cycle if v r+1 = v 1 Concordance number rotation number For a closed oriented polygonal plane curve C, let e and e be two consequent oriented segments of C, v their common vertex. Let l be an arbitrary oriented line. Define c l (e, e ) Z/2Z in the following way: c l (e, e ) = 1 if the directing vector of l belongs to the interior of the cone spanned by e and e, c l (e, e ) = 0 otherwise. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Planar Networks Boundary Measurements Paths and cycles A path P in N is an alternating sequence (v 1, e 1, v 2,..., e r, v r+1 ) of vertices and edges such that e i = (v i, v i+1 ) for any i [1, r]. A path is called a cycle if v r+1 = v 1 Concordance number rotation number For a closed oriented polygonal plane curve C, let e and e be two consequent oriented segments of C, v their common vertex. Let l be an arbitrary oriented line. Define c l (e, e ) Z/2Z in the following way: c l (e, e ) = 1 if the directing vector of l belongs to the interior of the cone spanned by e and e, c l (e, e ) = 0 otherwise. Define c l (C) as the sum of c l (e, e ) over all pairs of consequent segments in C. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Planar Networks Boundary Measurements Paths and cycles A path P in N is an alternating sequence (v 1, e 1, v 2,..., e r, v r+1 ) of vertices and edges such that e i = (v i, v i+1 ) for any i [1, r]. A path is called a cycle if v r+1 = v 1 Concordance number rotation number For a closed oriented polygonal plane curve C, let e and e be two consequent oriented segments of C, v their common vertex. Let l be an arbitrary oriented line. Define c l (e, e ) Z/2Z in the following way: c l (e, e ) = 1 if the directing vector of l belongs to the interior of the cone spanned by e and e, c l (e, e ) = 0 otherwise. Define c l (C) as the sum of c l (e, e ) over all pairs of consequent segments in C. c l (C) does not depend on l, provided l is not collinear to any of the segments in C. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Planar Networks Boundary Measurements Paths and cycles A path P in N is an alternating sequence (v 1, e 1, v 2,..., e r, v r+1 ) of vertices and edges such that e i = (v i, v i+1 ) for any i [1, r]. A path is called a cycle if v r+1 = v 1 Concordance number rotation number For a closed oriented polygonal plane curve C, let e and e be two consequent oriented segments of C, v their common vertex. Let l be an arbitrary oriented line. Define c l (e, e ) Z/2Z in the following way: c l (e, e ) = 1 if the directing vector of l belongs to the interior of the cone spanned by e and e, c l (e, e ) = 0 otherwise. Define c l (C) as the sum of c l (e, e ) over all pairs of consequent segments in C. c l (C) does not depend on l, provided l is not collinear to any of the segments in C. The common value of c l (C) for different choices of l is denoted by c(c) and called the concordance number of C. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 43 / 96

Example Planar Networks e 4 e8 e 7 e 5 e3 e 1 e 2 l e 6 Figure: c l (e 1, e 2 ) = c l (e 5, e 2 ) = 0; c l (e 2, e 3 ) = 1, c l (e 2, e 6 ) = 0; c l (e 6, e 7 ) = 1, c l (e 7, e 8 ) = 0 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 44 / 96

Planar Networks Boundary Measurements (cont d) Weight of a path A path P between a source b i and a sink b j (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 45 / 96

Planar Networks Boundary Measurements (cont d) Weight of a path A path P between a source b i and a sink b j a closed polygonal curve C P = P counterclockwise path btw. b j and b i along the boundary. The weight of P : w P = ( 1) c(c P) 1 e P w e. (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 45 / 96

Planar Networks Boundary Measurements (cont d) Weight of a path A path P between a source b i and a sink b j a closed polygonal curve C P = P counterclockwise path btw. b j and b i along the boundary. The weight of P : w P = ( 1) c(c P) 1 e P w e. Boundary Measurement M(i, j) = weights of all paths starting at b i and ending at b j (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 45 / 96