Poisson Geometry of Noncommutative Cluster Algebras Semeon - PowerPoint PPT Presentation

Poisson Geometry of Noncommutative Cluster Algebras Semeon Arthamonov UC Berkeley November 11, 2019 Canonical Bases, Cluster Structures and NC Birational Geometry AMS Fall Western Sectional Meeting, UC Riverside S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 1 / 32

Cluster algebras associated to triangulated surfaces Figure: Flip of an ideal triangulation corresponds to mutation. M. Gekhtman , M. Shapiro, and A. Vainshtein Cluster algebras and Weil-Petersson forms. Duke Mathematical Journal, 127(2), 291-311, 2005. V. Fock, A. Goncharov, A. Moduli spaces of local systems and higher Teichm¨ uller ematiques de l’IH ´ theory. Publications Math´ ES, 103, 1-211, 2006. S. Fomin, M. Shapiro, and D. Thurston. Cluster algebras and triangulated surfaces . Part I: Cluster complexes. Acta Mathematica, 201(1):83–146, 2007. A. Goncharov, and R. Kenyon. Dimers and cluster integrable systems . Annales scientifiques de l’ ´ Ecole Normale Sup´ erieure. Vol. 46. No. 5., 2013. S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 2 / 32

Ribbon Graphs Definition A ribbon graph Γ is a graph with cyclic order of edges adjacent to each vertex. (b) Disc in S Γ corresponding to (a) Ribbon Graph the vertex. Figure: Surface with boundary S Γ associated to a ribbon graph. Each ribbon graph Γ defines an oriented surface S Γ with boundary. S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 3 / 32

Ideal triangulations and bipartite graphs Figure: Bipartite ribbon graph associated to triangulation of surface Σ . Definition A conjugate surface ˆ S Γ associated to the bipartite ribbon graph Γ is a surface corresponding to the ribbon graph with reversed cyclic order of edges at each black vertex. Surface ˆ S Γ has the same fundamental group as the underlying graph π 1 ( ˆ S Γ ) = π 1 (Γ) . (1) S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 4 / 32

Graph connections and edge weights Fix arbitrary orientation of Γ and let G be a linear algebraic group. Definition Graph connection is an assignment of a parallel transport g e ∈ G to each oriented edge e ∈ E (Γ) . The gauge group G V (Γ) = { V (Γ) → G , v �→ h v } acts on a graph connections g e �→ h t ( e ) g e h − 1 s ( e ) , where s ( e ) and t ( e ) stand for the source and target of an edge e respectively. Conjugacy class of a parallel transport M F = g e k . . . g e 2 g e 1 M F along the edge loop is invariant under the action of G V (Γ) . Variables of a cluster chart in [G.-K.] correspond to G = GL ( 1 ) . S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 5 / 32

Cluster algebras and Poisson Geometry Let k be a ground field of characteristic zero. Definition Let A be a commutative associative algebra over k , a k -linear map { , } : A ⊗ A − → A is called a Poisson bracket on A if it satisfies for all f , g , h ∈ M { f , g } = −{ g , f } skew-symmetry condition, { f , gh } = g { f , h } + { f , g } h Leibnitz identity, { f , { g , h }} + { g , { h , f }} + { h , { f , g }} = 0 Jacobi identity, Geometric Cluster Algebras can be equipped with a Poisson bracket compatible with mutations. M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster algebras and Poisson geometry. Moscow Mathematical Journal, 3(3):899–934, 2003. S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 6 / 32

Poisson bracket on graph connections Each 1-dimensional representation ϕ ∈ Hom ( π 1 ( S Γ ) , C × ) is determined by y 1 = ϕ ( M 1 ) , . . . , y n = ϕ ( M n ) . We can equip C [ y 1 , . . . , y n ] with a Poisson bracket as follows � { y i , y j } = ǫ i , j ( p ) y i y j , p M 1 M 2  M j M i     + 1   p   M 3 ǫ ij ( p ) = M j M i     − 1   p   S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 7 / 32

Rectangle move y 1 y 4 z 1 z 4 y 0 z 0 y 2 y 3 z 2 z 3 Figure: Rectangle move in one dimensional case. Proposition (Goncharov-Kenyon’2013) The following map extends to a homomorphism of Poisson algebras  z 0 → y − 1 0 ,  τ : z i → y i ( 1 + y 0 ) , i = 1 , 3 ,  z i → y i ( 1 + y − 1 0 ) − 1 , i = 2 , 4 . S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 8 / 32

Noncommutative cluster algebras Cluster algebras with noncommutative edge weights A. Berenstein, V. Retakh Noncommutative marked surfaces. Advances in Mathematics, 328, 1010-1087, 2018. Can we equip them with a “Poisson bracket” compatible with mutations? Spoiler: Yes, but there are several challenges: Usual notion of a Poisson bracket is too restrictive. In every 1 noncommutative ring it forced to be a multiple of the commutator. 1 Can no longer use loops without a base point to define variables of a 2 cluster chart. Considering loops with base point will destroy “locality” of the mutation. 3 Solution: Introduce a noncommutative bi-vector field acting on open arcs. 1 for a proof, see D. Farkas and G. Letzter. Ring theory from symplectic geometry. Journal of Pure and Applied Algebra, 125(1–3):155 – 190, 1998. S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 9 / 32

Algebra of polyvector fields A vector field d on M can be viewed as the derivation of C ∞ ( M ) , the algebra of smooth functions on M d : C ∞ ( M ) → C ∞ ( M ) , d ( fg ) = fd ( g ) + d ( f ) g , for all f , g ∈ C ∞ ( M ) . Lemma The space of vector fields D 1 = Der ( C ∞ ( M ) , C ∞ ( M )) forms a C ∞ ( M ) -module. One defines an algebra of polyvector fields as D • = T C ∞ ( M ) D 1 . Puzzle: Der ( A , A ) is no longer an A -module for a noncommutative algebra A . S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 10 / 32

Double Geometry W. Crawley-Boevey, P . Etingof, and V. Ginzburg. Noncommutative geometry and quiver algebras . Advances in Mathematics, 209(1):274 – 336, 2007 M. Van den Bergh. Double Poisson algebras . Trans. Amer. Math. Soc., 360:5711–5769, 2008. Definition Let A be an associative algebra. We say that map δ is a noncommutative vector field if δ : A → A ⊗ A , δ ( ab ) = ( a ⊗ 1 ) δ ( b ) + δ ( a )( 1 ⊗ b ) for all a , b ∈ A . Lemma Noncommutative vector fields DA = Der ( A , A ⊗ A ) form an A -bimodule. One defines a noncommutative algebra of polyvector fields as T A DA S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 11 / 32

Double derivations of a ring with many objects Definition Let C be a small k -linear category. For all V , W ∈ Obj C we say that a map δ : Mor C → hom( W , − ) ⊗ hom( − , V ) is a ( V , W ) -vector field if δ ( f ◦ g ) = ( f ⊗ 1 V ) ◦ δ ( g ) + δ ( f ) ◦ ( 1 W ⊗ g ) . for all composable f , g ∈ Mor C . Here 1 V and 1 W are the identity morphisms on V and W . In what follows we denote the space of ( V , W ) -vector fields as D 1 V , W . Let ( a ⋆ δ ⋆ b )( f ) = ( δ ′ ( f ) ◦ b ) ⊗ ( a ◦ δ ′′ ( f )) (3) Lemma D 1 is a covariant functor on C × C op . S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 12 / 32

Modules over a ring with many objects Fix a small k -linear category C . Definition (Tensor product) Let R be a contrvariant functor on C and L be a covariant functor on C , the tensor product R ⊗ C L is defined as    � R ⊗ C L = R V ⊗ L V .  � V ∈ Obj C ρ ◦ f ⊗ λ ∼ ρ ⊗ f ◦ λ Definition (Trace) Let M be a bifunctor on C , the trace over C is defined as    � tr C : M → M ♮ := M X , X .  � X ∈ Obj C f ◦ m ∼ m ◦ f S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 13 / 32

Polyvector fields The space of k-vector fields associated to V , W ∈ Obj C is defined as � D k D 1 ⊗ C D 1 V , W = V , U 1 ⊗ C . . . U k − 1 , W , U 1 ,..., U k − 1 ∈ Obj C where for k = 0 we assume that D 0 V , W = hom( W , V ) . Corollary D k is a covariant functor on C × C op . ∞ � D k D • V , W = V , W k = 0 We define the category V of polyvector fields on C as hom V ( W , V ) = D • Obj V = Obj C , V , W . S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 14 / 32

Traces of polyvector fields and polyderivations Let δ 1 , . . . , δ k be a chain of composable vector fields. The trace tr C ( δ 1 ⋆ · · · ⋆ δ k ) is equivalent to the following map ( Mor C ) ⊗ k → ( Mor C ) ⊗ k , tr C ( δ 1 ⋆ · · · ⋆ δ k ) : ( δ ′ k ( f k ) ◦ δ ′′ 1 ( f 1 )) ⊗ ( δ ′ 1 ( f 1 ) ◦ δ ′′ f 1 ⊗ · · · ⊗ f k �→ 2 ( f 2 )) ⊗ . . . · · · ⊗ ( δ ′ k − 1 ( f k − 1 ) ◦ δ ′′ k ( f k )) . Proposition ∆ = tr C ( δ 1 ⋆ · · · ⋆ δ k ) is a polyderivation, i.e., ∆( h 1 ⊗ · · · ⊗ f ◦ g ⊗ · · · ⊗ h k ) ↑ j =( 1 t ( h k ) ⊗ · · · ⊗ f ⊗ · · · ⊗ 1 t ( h k − 1 ) ) ◦ ∆( h 1 ⊗ · · · ⊗ g ⊗ · · · ⊗ h k ) ↑ ↑ j + 1 j + ∆( h 1 ⊗ · · · ⊗ f ⊗ · · · ⊗ h k ) ◦ ( 1 s ( h 1 ) ⊗ · · · ⊗ g ⊗ · · · ⊗ 1 s ( h k ) ) ↑ ↑ j j S. Arthamonov Poisson Geometry of NC Cluster Algebras AMS Sectional Meeting, UC Riverside 15 / 32