Noncommutative Poisson Geometry and Cluster Integrable Systems S. Arthamonov Rutgers, The State University of New Jersey May 7, 2018 Cluster Algebras and Mathematical Physics S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 1 / 26
Cluster algebras associated to triangulated surfaces Figure: Flip of an ideal triangulation. 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. A. Berenstein and V. Retakh. Noncommutative marked surfaces. Advances in Mathematics, 328:1010–1087, 2018. S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 2 / 26
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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 3 / 26
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 (a) Ribbon Graph to 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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 4 / 26
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 ribbon graph Γ is a surface corresponding to the ribbon graph with reversed cyclic order of edges at each vertex. Both ˆ S Γ and S Γ have the same fundamental group as the underlying graph π 1 ( ˆ S Γ ) = π 1 ( S Γ ) = π 1 (Γ) . (1) The identification (1) allows one to introduce two different Poisson structures on the character variety of π 1 (Γ) . S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 5 / 26
Poisson bracket on graph connections Each 1-dimensional representation ϕ ∈ Hom ( π 1 ( S Γ ) , C × ) is determined by x 1 = ϕ ( M 1 ) , . . . , x n = ϕ ( M n ) . We can equip C [ x 1 , . . . , x n ] with a Poisson bracket as follows � ⊗ 2 → C [ x 1 , . . . , x n ] , � � { , } : C [ x 1 , . . . , x n ] { x i , x j } = ǫ i , j ( p ) x i x j , p M j M i + 1 p ǫ ij ( p ) = M j M i − 1 p S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 6 / 26
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 , z i → y i ( 1 + y 0 ) , i = 1 , 3 , τ : z i → y i ( 1 + y − 1 0 ) − 1 , i = 2 , 4 . S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 7 / 26
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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 8 / 26
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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 9 / 26
Vector fields on a linear category 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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 10 / 26
Modules over a linear category 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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 11 / 26
Category of 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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 12 / 26
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 NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 13 / 26
Double Quasi Poisson Bracket on a category Definition A k -linear map R is said to be a Double Quasi Poisson Bracket if it satisfies Skew-Symmetry condition � � op R ( f ⊗ g ) = − R ( g ⊗ f ) Double Leibnitz Identity R (( f ◦ g ) ⊗ h ) =( 1 t ( h ) ⊗ f ) ◦ R ( g ⊗ h ) + R ( f ⊗ h ) ◦ ( g ⊗ 1 s ( h ) ) , R ( f ⊗ ( g ◦ h )) =( g ⊗ 1 t ( f ) ) ◦ R ( f ⊗ h ) + R ( f ⊗ g ) ◦ ( 1 s ( f ) ⊗ h ) . Double Quasi Jacobi Identity � R 1 , 2 ◦ R 2 , 3 + R 2 , 3 ◦ R 3 , 1 + R 3 , 1 ◦ R 1 , 2 = tr C ( ∂ V ⋆ ∂ V ⋆ ∂ V ) . V ∈ Obj C R i , j ( f 1 ⊗ · · · ⊗ f n ) = f 1 ⊗ · · · ⊗ R ′ ( f i ⊗ f j ) ⊗ · · · ⊗ R ′′ ( f i ⊗ f j ) ⊗ · · · ⊗ f n . � �� � � �� � i j S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 14 / 26
Category associated to a ribbon graph C 0 = k π 1 ( ˆ S Γ , V 1 , . . . , V n ) . The objects Obj C 0 = { V i } correspond to marked points. x 3 f 2 f 1 x 1 f − 1 ◦ f − 1 x 2 1 2 (a) Disk corresponding to white vertex (b) Disc corresponding to black vertex Figure: Building blocks for bipartite graph with trivalent black vertices S. Arthamonov NC Poisson Geometry and Cluster Algebras CAMP-2018, East Lansing MI 15 / 26
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