Representation homology and derived character maps Sasha Patotski Cornell University ap744@cornell.edu April 30, 2016 Sasha Patotski (Cornell University) Representation homology April 30, 2016 1 / 18
Plan 1 Classical representation schemes. 2 Derived representation schemes and representation homology. 3 Derived character maps. Sasha Patotski (Cornell University) Representation homology April 30, 2016 2 / 18
Representation schemes Assumption: k is a fixed field of char ( k ) = 0, all algebras are over k , ⊗ denotes ⊗ k . A graded algebra B is commutative if for a , b ∈ B ab = ( − 1) deg( a ) deg( b ) ba Let A ∈ Alg k be an associative algebra, V = k n an n -dimensional vector space. By Rep n ( A ) we denote the moduli space of representations of A in k n . ≃ A rn 2 . Example. Rep n ( k � x 1 , . . . , x r � ) = Mat × r n Example. Rep n ( k [ x 1 , . . . , x r ]) ⊂ Mat × r is the closed subscheme, n consisting of tuples ( B 1 , . . . , B r ) of pair-wise commuting matrices. Sasha Patotski (Cornell University) Representation homology April 30, 2016 3 / 18
� �� � � Character map Characters define a linear map Tr: A → k [ Rep n ( A )] a �→ [Tr( a ): ρ �→ tr( ρ ( a ))] , ∀ ρ ∈ Rep n ( A ) This map factors as Tr � k [ Rep n ( A )] A i � k [ Rep n ( A )] GL n A / [ A , A ] The map A / [ A , A ] → k [ Rep n ( A )] GL n will be called the character map . Theorem (Procesi) The induced map Sym(Tr): Sym( A / [ A , A ]) → k [ Rep n ( A )] GL n is surjective. Sasha Patotski (Cornell University) Representation homology April 30, 2016 4 / 18
Extension to DG algebras In general, Rep n ( A ) is “badly behaved,” for example, it is quite singular even for “nice” algebras (e.g. A = k [ x 1 , ..., x d ] , d > 1) Solution: “resolve singularities” by deriving Rep n . Call the functor ( − ) n : Alg k → ComAlg k sending A �→ A n := k [ Rep n ( A )] the representation functor . It extends naturally to ( − ) n : DGA k → CDGA k . Problem: The functor ( − ) n is not “exact”, i.e. it does not preserve quasi-isomorphisms. Sasha Patotski (Cornell University) Representation homology April 30, 2016 5 / 18
Derived representation functor Theorem (Berest–Khachatryan–Ramadoss) The functor ( − ) n has a total left derived functor L ( − ) n computed by ∼ L ( A ) n = R n for any resolution R ։ A. The algebra L ( A ) n does not depend on the choice of resolution, up to quasi-isomorphism. For A ∈ Alg k , a resolution is any semi-free DG algebra R ∈ DGA k with a ∼ surjective quasi-isomorphism R ։ A . Denote L A n by DRep n ( A ), call it derived representation scheme . Example: If A = k [ x , y ], take R = k � x , y , λ � with deg( x ) = deg( y ) = 0, deg( λ ) = 1 and d λ = xy − yx . Then DRep n ( A ) = k [ x ij , y ij , λ ij ] with deg( λ ij ) = 1 and n � d λ ij = x ik y kj − y ik x kj k =1 Sasha Patotski (Cornell University) Representation homology April 30, 2016 6 / 18
Representation homology Define n -dimensional representation homology by H • ( A , n ) := H • [DRep n ( A )] Facts: 1 H 0 ( A , n ) ≃ k [ Rep n ( A )] =: A n . 2 If Rep n ( A ) = ∅ , then H • ( A , n ) = 0. 3 for A formally smooth, H p ( A , n ) = 0 for ∀ n ≥ 1 and p ≥ 1. ∼ 4 DRep 1 ( A ) ≃ R ab for any resolution R ։ A , so H • ( A , 1) ≃ H • ( R ab ) Sasha Patotski (Cornell University) Representation homology April 30, 2016 7 / 18
Example: polynomial algebra on two variables Let A = k [ x , y ], R = k � x , y , λ � with d λ = xy − yx . Then DRep 1 ( A ) ≃ k [ x , y , λ ] with zero differential, so H • ( k [ x , y ] , 1) ≃ k [ x , y ] ⊕ k [ x , y ] .λ � �� � � �� � deg=0 deg=1 H • ( k [ x , y ] , 2) ≃ k [ x , y ] 2 ⊗ Sym ( ξ, τ, η ) / I with ξ, τ, η of degree 1 and I the ideal generated by the relations x 12 η − y 12 ξ = ( x 12 y 11 − y 12 x 11 ) τ x 21 η − y 21 ξ = ( x 21 y 22 − y 21 x 22 ) τ ( x 11 − x 22 ) η − ( y 11 − y 22 ) ξ = ( x 11 y 22 − y 11 x 22 ) τ ξη = y 11 ξτ − x 11 ητ = y 22 ξτ − x 22 ητ Theorem (Berest-Felder-Ramadoss) For i > n we have H i ( k [ x , y ] , n ) = 0 . Sasha Patotski (Cornell University) Representation homology April 30, 2016 8 / 18
Example: q-polynomials and dual numbers Let q ∈ k × , and define k q [ x , y ] = k � x , y � / ( xy = qyx ). Theorem (Berest–Felder–Ramadoss) If q is not a root of 1 , then for all n ≥ 1 H p ( k q [ x , y ] , n ) = 0 , ∀ p > 0 For A = k [ x ] / ( x 2 ) the minimal resolution is R = k � t 0 , t 1 , t 2 , . . . � with deg t i = i and dt p = t 0 t p − 1 − t 1 t p − 2 + · · · + ( − 1) p − 1 t p − 1 t 0 In this case even for H • ( A , 1) = H • ( R ab ) don’t have a good description. Sasha Patotski (Cornell University) Representation homology April 30, 2016 9 / 18
Relation to Lie homology Let C be a (augmented) DG coalgebra Koszul dual to A ∈ Alg k ∼ (augmented), i.e. Ω( C ) → A . Theorem (Berest–Felder–P–Ramadoss–Willwacher) There is an isomorphism H • ( A , n ) GL n ≃ H • ( gl ∗ n ( ¯ H • ( A , n ) ≃ H • ( gl ∗ n ( C ) , gl ∗ C ); k ) , n ( k ); k ) If dim( C ) < ∞ , take E = C ∗ the linear dual DG algebra. Then H • ( A , n ) GL n ≃ H −• ( gl n ( E ) , gl n ( k ); k ) H • ( A , n ) ≃ H −• ( gl n ( ¯ E ); k ) , Sasha Patotski (Cornell University) Representation homology April 30, 2016 10 / 18
Derived character maps Want: relate H • ( A , n ) to more computable invariants. Proposition (Berest-Khachatryan-Ramadoss) For any algebra A ∈ Alg k and any n there exists a canonical derived character map Tr n ( A ) • : HC • ( A ) → H • ( A , n ) GL n , extending the original character map Tr: HC 0 ( A ) = A / [ A , A ] → A GL n n Sasha Patotski (Cornell University) Representation homology April 30, 2016 11 / 18
Symmetric algebras Goal: compute derived character maps for A = Sym( W ). For simplicity, assume n = 1 (i.e. only consider H • ( A , 1)). Tr( A ) • factors through the reduced cyclic homology HC • ( A ). For A = Sym( W ), HC i ( A ) ≃ Ω i ( W ) / d Ω i − 1 ( W ), Ω i ( W ) ≃ Sym( W ) ⊗ Λ i ( W ) Thus, we can think of Tr( A ) i as maps Tr( A ) i : Ω i ( W ) → H i ( A , 1) Sasha Patotski (Cornell University) Representation homology April 30, 2016 12 / 18
Example: A = k [ x , y ] DRep 1 ( k [ x , y ]) ≃ k [ x , y , λ ] with zero differential. The character Tr 0 : k [ x , y ] → k [ x , y , λ ] is given by Tr 0 ( P ( x , y )) = P ( x , y ) for any P ( x , y ) ∈ k [ x , y ]. The character Tr 1 : Ω 1 ( A ) → k [ x , y , λ ] is given by � ∂ Q � ∂ x − ∂ P Tr 1 ( P ( x , y ) dx + Q ( x , y ) dy ) = λ ∂ y Sasha Patotski (Cornell University) Representation homology April 30, 2016 13 / 18
Tr ( A ) 1 for A = Sym( W ) For A = Sym( W ) ≃ k [ x 1 , . . . , x m ] Λ 2 ( W ) ⊕ · · · ⊕ Λ m ( W ) DRep 1 ( A ) ≃ Sym( W ) ⊗ Sym . � �� � � �� � deg=1 deg= m − 1 with zero differential , so H • ( A , 1) ≃ DRep 1 ( A ). λ ( v 1 , v 2 , . . . , v p ) := v 1 ∧ v 2 ∧ . . . ∧ v p ∈ Λ p ( W ) ⊂ DRep 1 ( A ) � �� � deg= p − 1 Proposition For α = � P i dx i ∈ Ω 1 ( A ) the map Tr( A ) 1 is given by � ∂ P i � � − ∂ P j Tr( A ) 1 ( α ) = λ ( x i , x j ) ∈ H • ( A , 1) ∂ x j ∂ x i i < j Sasha Patotski (Cornell University) Representation homology April 30, 2016 14 / 18
Example: Tr 2 for A = k [ x , y , z ] Take ω = Pdx ∧ dy + Qdy ∧ dz + Rdz ∧ dx ∈ Ω 2 ( A ). Then Tr( A ) 2 ( ω ) is given by M λ ( x , y , z )+ M y λ ( x , y ) λ ( y , z )+ M z λ ( y , z ) λ ( z , x )+ M x λ ( z , x ) λ ( x , y ), where M := P z + Q x + R y and for a polynomial F , F q denotes ∂ F ∂ q . Tr( A ) 2 = D ◦ d dR , where D : Ω 3 → H • ( A , 1) D = s − 1 + � is a certain canonical differential operator on differential forms. Sasha Patotski (Cornell University) Representation homology April 30, 2016 15 / 18
Abstract Chern–Simons forms Let A be a cohomologically graded commutative DG algebra, g a finite dimensional Lie algebra. A g -valued connection is an element θ ∈ A 1 ⊗ g . Its curvature is Θ := d θ + 1 2 [ θ, θ ], and Bianchi identity holds: d Θ = [Θ , θ ] If P ∈ k [ g ] ad g , deg( P ) = r , for α ∈ A ⊗ Sym r ( g ) define P ( α ) ∈ A via 1 r ! id ⊗ ev P � A A ⊗ Sym r ( g ) Then P (Θ r ) ∈ A 2 r is exact, and there exists CS P ( θ ) ∈ A 2 r − 1 such that d CS P ( θ ) = P (Θ r ) with CS P ( θ ) is given explicitly by 1 � P ( θ ∧ Θ r − 1 CS P ( θ ) = r ! ) dt t 0 2 ( t 2 − t )[ θ, θ ]. where Θ t = t Θ + 1 Sasha Patotski (Cornell University) Representation homology April 30, 2016 16 / 18
Derived character maps for polynomial algebras Take A = hom(Ω • ( W ) , R ab ), g = k , and P r = x r ∈ k [ g ] ≃ k [ x ]. Theorem (Berest-Felder-P-Ramadoss-Willwacher) There is a canonical k-valued connection θ in A such that the derived character map Tr( A ) • : Ω • ( A ) → R ab ≃ H • ( A , 1) is given by � ∞ Tr( A ) • = CS P r ( θ ) ◦ d. r =0 Here, θ ( P ( x 1 , . . . , x m ) dx i 1 . . . dx i p ) = P (0 , . . . , 0) λ ( x i 1 , . . . , x i p ) Remark: this allows to get explicit formulas for all derived character maps. Sasha Patotski (Cornell University) Representation homology April 30, 2016 17 / 18
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