Cohomological Hall algebra of a preprojective algebra

Transcription

1 Cohomological Hall algebra of a preprojective algebra Gufang Zhao Institut de Mathématiques de Jussieu-Paris Rive Gauche Conference on Representation Theory and Commutative Algebra Apr. 24, 2015, Storrs, CT Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 0 / 16

2 Outline 1 Cohomological Hall algebras Classical Hall algebra Cohomological Hall algebra 2 CoHA of preprojective algebras Oriented cohomology theories Hall multiplication Shuffle algebra 3 Representations 4 Quantum affine algebras Yangian and quantum loop algebra Future works Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 0 / 16

3 Cohomological Hall algebras Classical Hall algebra Hall algebra of a quiver Let Q = (I, H) be a quiver. Ringel defined the Hall algebra: with multiplication H(Q) = Z{[M] : iso. class of repns. of Q over F q }, [M] [N] = g X M,N [X] where g X = #{U X U M, X/U N}. M,N Theorem (Ringel, 1990) For any Q, H(Q) is an associative algebra. If Q is of ADE type, then there is an isomorphism X H(Q) U q (n + ) U q (g(q)). Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 1 / 16

4 Cohomological Hall algebras Cohomological Hall algebra Cohomological Hall algebra of a quiver For v N I, the group G v = i I GL vi acts on Rep(Q, v). Theorem (Kontsevich-Soibelman, 2008) H G (Rep(Q)) := v N IH G v (Rep(Q, v)) is an associative algebra. For v 1, v 2 N I dim. vectors, v = v 1 + v 2. Let C v be a vector space with dim. vector v, and C v 1 C v be the subspace with dim. vector v 1. Denote Y = Rep(Q, v 1 ) Rep(Q, v 2 ). Y V Rep(Q, v), where V = {x Rep(Q, v) x(c v 1) C v 1} Rep(Q, v). P = G v1,v 2 G v be the parabolic subgroup preserving C v 1. G v P Y φ G v P V ψ Rep(Q, v) is G v -equivariant. The multiplication is defined to be m = ψ φ. Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 2 / 16

5 Cohomological Hall algebras Cohomological Hall algebra Main goal Study the cohomological Hall algebra of preprojective algebra of Q. Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 3 / 16

6 Table of Contents CoHA of preprojective algebras 1 Cohomological Hall algebras Classical Hall algebra Cohomological Hall algebra 2 CoHA of preprojective algebras Oriented cohomology theories Hall multiplication Shuffle algebra 3 Representations 4 Quantum affine algebras Yangian and quantum loop algebra Future works Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 3 / 16

7 CoHA of preprojective algebras Oriented cohomology theories Oriented cohomology theories Definition (Levine-Morel 2007) An oriented cohomology theory (OTC) on Sm k is D1 A functor A : Sm op Comm. Rings. k D2 For any projective morphism f : Y X in Sm k, f : A (Y) A (X), a morphism of A (X)-modules. f is functorial w.r.t. proj. morphisms. These data satisfy: transversal base change, the projective bundle formula; and extended homotopy property. For L X a line bundle, the first Chern class is defined as c A 1 (L) := s s (1 X ), where s : X L is the zero section. For any linear algebraic group G, using Borel construction one extend A to A G by A G (X) := A (X G EG). Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 4 / 16

8 CoHA of preprojective algebras Oriented cohomology theories Formal group laws Let A be an OCT, then there is a power series F(u, v) A (pt)[[u, v]] such that for any X and line bundle L, M on X, we have c 1 (L M) = F(c 1 (L), c 1 (M)) A (X). The series F(u, v) is a formal group law (FGL) satisfing F(x, y) = F(y, x), F(x, 0) = x, F(x, F(y, z)) = F(F(x, y), z). We will denotex + F y = F(x, y), and denote F X to be the inverse of X under this formal group law. Example, let E be an elliptic curve, with local coordinate l. Then F(l(u), l(v)) = l(u + v) defines an elliptic FGL. The corresponding OCT is called an elliptic cohomology. Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 5 / 16

9 CoHA of preprojective algebras Hall multiplication Main questions The group G v and the torus T = G m 2 act on T Rep(Q, v). Let µ v : T Rep(Q, v) gl v be the moment map. µ 1 v (0) with the G v -action is the representation space of the preprojective algbera of Q. Question Can one define a Hall multiplication on v N IA T G v (µ 1 v (0))? Can one construct representations of this algebra out of Nakajima quiver varieties (moduli spaces of stable framed representations of the preprojective algebra)? What is the relation between this action and the actions of quantum affine algebras constructed by Nakajima? Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 6 / 16

10 CoHA of preprojective algebras Hall multiplication Define A T G v (µ 1 v (0)) := A (T T G v Rep(Q, v), T Rep(Q, v) \ µ 1 v (0)). Theorem (Yang-Z., 2014) P A (Q) := v N IA T G v (µ 1 v (0)) is an associative algebra. For v 1, v 2 N I and v = v 1 + v 2, recall φ X := G v P Y W := G v P V Rep(Q, v). Here Y = Rep(Q, v 1 ) Rep(Q, v 2 ) and V = {x Rep(Q, v) x(c v 1) C v 1} Rep(Q, v). Z := T (X Rep(Q, v)) the conormal bundle of W X Rep(Q, v). W We have the following correspondence of G v T-varieties: G P T L Y Z G T G G P T Y ι T G X T X The multiplication is m v1,v 2 = ψ φ ι. Φ Z Ψ ψ Rep(Q, v) T Rep(Q, v) Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 7 / 16

11 CoHA of preprojective algebras Shuffle algebra Jordan quiver case Let Q be the Jordan quiver, K be the K-theory. The algebra P K (Q) has been studied by Schiffmann-Vasserot. Theorem (Schiffmann-Vasserot, 2008) There is an action of P K (Q) on n K T (Hilb n (C 2 )). There is an algebra map P K (Q) to the shuffle algebra S. Here S = r N S r with S r = Q[t ± 1, t± 2 ][z± 1,..., z± r ] S r. And multiplication S r Q[t ± 1,t± 2 ] S r S r+r f(z 1,..., z r ) g(z 1,..., z r ) = σ(h f(z 1,..., z r )g(z r+1,..., z r+r )) where h = i [1,r] j [r+1,r+r ] σ Sh(r,r ) (1 t 1 z i /z j )(1 t 2 z i /z j ) (1 z i /z j )(1 t 1 t 2 z i /z j ). Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 8 / 16

12 CoHA of preprojective algebras Shuffle algebra The formal shuffle algebras Let Q be any quiver. Let A be an OCT whose FGL is (R, F). Let SH F (Q) = v N ISH v with SH v := R[[t 1, t 2 ]][[λ i s]] S v i I,s=1,...,v i. For any v 1 and v 2 N I. The multiplication SH v1 R[[t1,t 2 ]] SH v2 SH v1 +v 2 has formula f 1 (λ ) f 2 (λ ) σ Sh(v 1,v 2 ) where fac 1 = v i v i 1 2 λ i t F λ i s+ F t 1 + F t 2 i I s=1 t=1 λ i t F λ i s σ(fac 1 fac 2 f 1 f 2 ). and fac 2 = v i v j 1 2 i,j I s=1 t=1 (λ j t F λ i s + F t 1 ) a ij(λ j t F λ i s + F t 2 ) a ji. Theorem (Yang-Z., 2014) There is an algebra homomorphism Θ : P A (Q) SH F (Q). Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 9 / 16

13 Representations Table of Contents 1 Cohomological Hall algebras Classical Hall algebra Cohomological Hall algebra 2 CoHA of preprojective algebras Oriented cohomology theories Hall multiplication Shuffle algebra 3 Representations 4 Quantum affine algebras Yangian and quantum loop algebra Future works Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 9 / 16

14 Representations Nakajima quiver varieties Let Q be the framed quiver. For v, w N I, define Rep(Q, v, w) = Rep(Q, (v, w)). Let µ v,w : T Rep(Q, v, w) gl v be the moment map. Define M(v, w) = µ 1 v,w(0)//g v for suitable stability condition. Example Barth: When Q is the Jordan quiver, M(v, 1) Hilb v (C 2 ). Kraft-Procesi: When Q is of type A n, w = (d, 0,..., 0), then v N IM(v, w) is the cotangent bundle of the variety of n-step flags in C d. Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 10 / 16

15 Representations Nakajima raising operators Let e k be the dim. vector, valued 1 on vertex k I, 0 otherwise. There is a Lagrangian C + k (v, w) M(v, w) M(v + e k, w) and bundle L k. Convolution with c 1 (L k ) defines an operator c 1 (L k ) : A T Gw (M(v, w)) A T Gw (M(v + e k, w)). Theorem (Yang-Z., 2014) w N I, Φ : P A (Q) End( v N IA T Gw (M(v, w))). Let ξ ek be the natural representation of G ek, and c 1 (ξ ek ) A T Gek (µ 1 e k (0)). Then Φ(c 1 (ξ ek )) = c 1 (L k ). Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 11 / 16

16 Table of Contents Quantum affine algebras 1 Cohomological Hall algebras Classical Hall algebra Cohomological Hall algebra 2 CoHA of preprojective algebras Oriented cohomology theories Hall multiplication Shuffle algebra 3 Representations 4 Quantum affine algebras Yangian and quantum loop algebra Future works Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 11 / 16

17 Quantum affine algebras Yangian and quantum loop algebra Quantum affine algebras Let Q be the quiver without edge-loop, and let g be the corresponding Lie algebra. Let Y (Lg) be the Yangian. It is a Hopf algebra, deforming U(g[z]). Let U q (Lg) be the quantum loop algebra. It is a Hopf algebra, deforming U(g[z, z 1 ]). Theorem (Nakajima 1999, Varagnolo 2000) 1 The Yangian Y (Lg) acts on v N IH G w G m (M(v, w)); 2 The quantum loop algebra U q (Lg) acts on v N IK Gw G m (M(v, w)). These are highest weight integrable representations. Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 12 / 16

18 Quantum affine algebras Yangian and quantum loop algebra Quantum affine algebras Let SP A (Q) P A (Q) be the subalgebra generated by A T Gek (µ 1 e k (0)) for k I. Corollary The following diagrams commute. Y + (Lg) Y (Lg) SP H t1 =0 (Q) t 2 = U + q (Lg) Φ End(H G w G m (M(w))) U q (Lg) SP K (Q) t 1 =0 t 2 =1 q 1 Φ End(K Gw G m (M(w))) Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 13 / 16

19 Quantum affine algebras Future works Future works: elliptic quantum group Let E τ, (sl 2 ) be the elliptic quantum group of sl 2 defined by Enriquez-Felder-Tarasov-Varchenko. Let Ell be the equivariant elliptic cohomology of G Ginzburg-Kapranov-Vasserot. Theorem (Z., to appear) Let Q be the quiver A 1. 1 There is an algebra isomorphism SP Ell t1 =0 (Q) E + τ, (sl 2). t 2 = 2 For any w > 0, E τ, (sl 2 ) acts on 0 v w Ell G w T (T Gr(v, w)) such that the following diagram commute E + τ, (sl 2) SP Ell t1 =0 (Q) Φ t 2 = E τ, (sl 2 ) End( 0 v w Ell G w T (T Gr(v, w))) Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 14 / 16

20 Quantum affine algebras Future works Future works: quiver with potentials Let Q be a quiver. Let W be a potential, studied by Derksen-Weyman-Zelevinsky. There is a CoHA for (Q, W) defined by Kontsevich-Soibelman. Can one construct representations of this CoHA from stable framed representations of (Q, W)? Gufang Zhao (IMJ-PRG) CoHA of preprojective algebra Apr. 24, 2015, Storrs, CT 15 / 16

21 Thank You!!!