Representation homology and derived character maps

Transcription

1 Representation homology and derived character maps Sasha Patotski Cornell University April 30, 2016 Sasha Patotski (Cornell University) Representation homology April 30, / 18

2 Plan 1 Classical representation schemes. 2 Derived representation schemes and representation homology. 3 Derived character maps. Sasha Patotski (Cornell University) Representation homology April 30, / 18

3 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. Example. Rep n (k x 1,..., x r ) = Mat r n A rn2. Example. Rep n (k[x 1,..., x r ]) Mat r n is the closed subscheme, consisting of tuples (B 1,..., B r ) of pair-wise commuting matrices. Sasha Patotski (Cornell University) Representation homology April 30, / 18

4 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 A A/[A, A] Tr k[rep n (A)] i k[rep n (A)] GLn The map A/[A, A] k[rep n (A)] GLn map. will be called the character Theorem (Procesi) The induced map Sym(Tr): Sym(A/[A, A]) k[rep n (A)] GLn is surjective. Sasha Patotski (Cornell University) Representation homology April 30, / 18

5 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 the representation functor. A A n := k[rep n (A)] 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, / 18

6 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 LA 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 dλ = xy yx. Then DRep n (A) = k[x ij, y ij, λ ij ] with deg(λ ij ) = 1 and dλ ij = n x ik y kj y ik x kj k=1 Sasha Patotski (Cornell University) Representation homology April 30, / 18

7 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, / 18

8 Example: polynomial algebra on two variables Let A = k[x, y], R = k x, y, λ with dλ = 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, / 18

9 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 ( 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, / 18

10 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) H (gl n( C); k), H (A, n) GLn H (gl n(c), gl n(k); k) If dim(c) <, take E = C the linear dual DG algebra. Then H (A, n) H (gl n (Ē); k), H (A, n) GLn H (gl n (E), gl n (k); k) Sasha Patotski (Cornell University) Representation homology April 30, / 18

11 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) GLn, extending the original character map Tr: HC 0 (A) = A/[A, A] A GLn n Sasha Patotski (Cornell University) Representation homology April 30, / 18

12 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 )/dω 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, / 18

13 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 for any P(x, y) k[x, y]. Tr 0 (P(x, y)) = P(x, y) The character Tr 1 : Ω 1 (A) k[x, y, λ] is given by Tr 1 (P(x, y)dx + Q(x, y)dy) = ( Q x P y ) λ Sasha Patotski (Cornell University) Representation homology April 30, / 18

14 Tr(A) 1 for A = Sym(W ) For A = Sym(W ) k[x 1,..., x m ] DRep 1 (A) Sym(W ) Sym Λ 2 (W ) Λ m (W ). }{{}}{{} deg=1 deg=m 1 with zero differential, so H (A, 1) DRep 1 (A). Proposition λ(v 1, v 2,..., v p ) := v 1 v 2... v p Λ p (W ) DRep }{{} 1 (A) deg=p 1 For α = P i dx i Ω 1 (A) the map Tr(A) 1 is given by Tr(A) 1 (α) = ( Pi P ) j λ(x i, x j ) H (A, 1) x j x i i<j Sasha Patotski (Cornell University) Representation homology April 30, / 18

15 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 Mλ(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 = s 1 + D : Ω 3 H (A, 1) is a certain canonical differential operator on differential forms. Sasha Patotski (Cornell University) Representation homology April 30, / 18

16 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 := dθ [θ, θ], and Bianchi identity holds: dθ = [Θ, θ] If P k[g] adg, deg(p) = r, for α A Sym r (g) define P(α) A via A Sym r (g) 1 r! id ev P A Then P(Θ r ) A 2r is exact, and there exists CS P (θ) A 2r 1 such that d CS P (θ) = P(Θ r ) with CS P (θ) is given explicitly by 1 CS P (θ) = r! where Θ t = tθ (t2 t)[θ, θ]. 0 P(θ Θ r 1 t )dt Sasha Patotski (Cornell University) Representation homology April 30, / 18

17 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 Pr (θ) d. r=0 Here, θ(p(x 1,..., x m )dx i1... dx ip ) = P(0,..., 0)λ(x i1,..., x ip ) Remark: this allows to get explicit formulas for all derived character maps. Sasha Patotski (Cornell University) Representation homology April 30, / 18

18 References Yu. Berest, G. Felder and A. Ramadoss, Derived representation schemes and noncommuative geometry, Expository lectures on representation theory, , Contemp. Math., 607, Amer. Math. Soc., Providence, RI, Yu. Berest, G. Khachatryan and A. Ramadoss, Derived representation schemes and cyclic homology, Adv. Math. 245 (2013), Yu. Berest, G. Felder, S. Patotski, A. Ramadoss, and Th. Willwacher Chern-Simons forms and Higher character maps of Lie representations, preprint. Yu. Berest, G. Felder, S. Patotski, A. Ramadoss, and Th. Willwacher Representation Homology, Lie Algebra Cohomology and Derived Harish-Chandra Homomorphism, preprint. Sasha Patotski (Cornell University) Representation homology April 30, / 18