R-matrices, affine quantum groups and applications

R-matrices, affine quantum groups and applications From a mini-course given by David Hernandez at Erwin Schrödinger Institute in January 2017 Abstract R-matrices are solutions of the quantum Yang-Baxter equation At the origin of the theory of quantum groups, they can be interpreted as intertwining operators in representation theory After reviewing standard constructions from quantum affine algebras, we will present recent development in the theory Maulik-Okounkov gave a general geometric construction of stable basis and R-matrices In another direction, monoidal categorifications of cluster algebras have been established using R-matrices to categorify Fomin-Zelevinsky mutations relations We also discuss recent advances on transfer matrices derived from R-matrices, which give new informations on corresponding quantum integrable systems as well as on the ODE/IM correspondence seen in the context of affine opers Contents 1 Algebraic solution (Drinfeld) 1 2 Geometric construction (theory stable envelop, Maulik-Okounkov 3 3 R matrices and categorification 5 4 Quantum integrable system 6 1 Algebraic solution (Drinfeld) Définition 11 Let A be an algebra over C and z be an indeterminate The Yang-Baxter equation (with spectral parameters) is as follows: R 12 (z)r 13 (zw)r 23 (w) = R 23 (w)r 13 (zw)r 12 (z), where R(z) (A A)(z), R 12 (z) = R(z) 1, R 23 (z) = 1 R(z) and R 13 (z) = (P Id)R 23 (z); P is a "twist" Exemple 12 A fundamental (historical) example (Yang R-matrix): A = End(C 2 ), q C, 1 0 0 0 0 q 1 (z 1) 1 q 2 0 z q R(z) = 2 z q 2 0 z(1 q 2 ) (z 1)q 1 0 z q 2 z q 1 0 0 0 1 1

It is attached to the 6-vertex model (ice) of XXZ-spin chain model If one writes z = e u, q = e h 2, then ( ) q 1 (z 1) ) z q 2 z(1 q 2 z q 2 1 q 2 z q 2 (z 1)q 1 z q 1 h,u 0 1 u + h ( u h h u Algebraic construction: V is a 2 dimensional representation of U q ( ˆ sl 2 ) (affine quantum group) We can generalize by replacing sl 2 by g a general simple finite dimensional Lie algebra and V by a finite dimensional representation of U q (ĝ) This gives rise to many R-matrices By Drinfeld-Jimbo, we have the following: g quantization U q (g) affinization quantization affine Kac Moody ĝ = Cc C[t ±1 ] g U q (ĝ) Theorem 13 (Drinfeld)"Commutative diagram" gives 2 presentation of U q (ĝ) Remark 14 U q (ĝ) is a Hopf algebra Theorem 15 U q (ĝ) has a universal non trivial R-matrix: R(z) (U q (ĝ) ˆ U q (ĝ)[[z]], where ˆ is some completion of the tensor product for some filtration, such that from two finite dimensional representations V, W of U q (ĝ), we get P R V,W (z) : V W [[z]] W V [[z]], where R V,W (z) is the image of R(z) in End(V W )[[z]]: we get a well defined morphism of U q (ĝ)-modules Proposition 16 If V, W are simple and finite dimensional, there exists f V,W C((z)) such that f V,W (z)r V,W (z) is rational in End((V W )) C(z) Let d Z be the order of 1 as a pole of R V,W (z)f V,W (z) Then lim z 1 [(z 1) d P f V,W (z)r V,W (z)] := R norm V,W : V W W V is a non-zero U q (ĝ)-module morphism Remark 17 In general we do not know the orders of poles and the zeros of rational points Exemple 18 see example above, g = sl 2, V = V (1), W = W (q 2 ), evaluation representation of U q ( sl ˆ 2 ) (morphism of algebra U q ( sl ˆ 2 ) U q ( sl ˆ 2 )) Then d = 1, ie 1 is a pole of order 1 and has rank 1 (not invertible) R norm V,W Remark 19 If U, V, W are finite dimensional representations of U q (ĝ): U V W P R U,V (f) Id V U Id P W R U,W (zw) V W U Id P R V,W (w) U W V W U V W V U By the Yang-Baxter equation, this diagram is commutative 2

2 Geometric construction (theory stable envelop, Maulik- Okounkov We consider the couple (X, ω), where X is a quasi-projective, non singular algebraic variety and ω is a holomorphic simplectic form We have an action of (C ) R+n T A (C ) R Hypothesis: Cω H 0 (Ω 2 ω) is stable for the induced action of T We have a corresponding character h : T C and A Kerh there exists a proper T -equivariant affine map F : X X 0 satisfying an additional technical formality condition Exemple 21 A = (C ) n+1, T = A C, X = T P n (action on coordinates of P n ) An additional functor acts on the fibers Nakajima quiver varieties Let X A be the set of fixed points for the action of A, and N(X A ) be the normal bundle X A It has a structure of a direct sum of 1 dimensional representations of A Définition 22 Let be the set of characters of A corresponding to A-simple modules occurring in N(X A ) One has X R = C(A) Z R, where C(A) is the group of characters of A One has X R R n Lie(A) For α, we have the hyperplane α = {v X R α(v) = 0} We have the decomposition X R \ α α = i C i, where the C i are the open chambers For a connected component Z X A and a chamber C, we set leaf C (Z) = {x X lim z 0 (σ(z)x) exists and is in Z} for some σ C (this does not depend on the choice of σ) For each chamber, we define a partial ordering C on the connected components of X A by saying Z C Z if leaf C Z Z The slope of a connected component Z is then Slope C (Z) = Z Z leaf C(Z ) Exemple 23 X = T P 1, = {α, α}, α(a, b) = a u 0 b u 1 where u 0, u 1 are characters for the action on A One has X R = R 2 = C + α C, where C ɛ = {(x, y) R 2 ɛu 0 x > u 1 y} for ɛ {, +} Let p 0 = [1 : 0], p 1 = [0 : 1] One has P 1 T P 1 One has leaf C+ ({p 0 }) = Tp 0 (P 1 ), leaf C ({p 1 }) = Tp 1 (P 1 ), leaf C+ ({p 1 }) = P 1 \{p 1 }, leaf C+ ({p 0 }) = P 1 \{p 0 } Let H T (X), H T (XA ) be the T -equivariant cohomology Theorem 24 (Maulik-Okounkov) There exists a unique Stab C : H T (XA ) H T (X) morphism of H T ({pt})-modules such that forγ H T (Z), with Z a connected component of XA and µ = Stab C (γ), we have : 1 Supp(µ) Slope C (Z) 3

2 µ Z = ±e(n (Z)) γ where N (Z) is the "negative part of µ(z) with respect to C" (the sign is decided with the choice of a "polarization") 3 for Z C Z, deg A (µ Z ) < codim Z (Z ) Idea of construction: based on the construction of a Lagrangian correspondence such that p 1 is proper Then Stab C = (p 1 ) (p 2 ) X A L X X A p 2 Exemple 25 X = T (P 1 ), u 0, u 1, h characters, H T ({pt}) = C[u 0, u 1, h], H T (X) = C[u 0, u 1, h] C[u 0, u 1, h]c, where c is the first Cherr character of O( 1) H T (XA ) = C[u 0, u 1, h][p 0 ] C[u 0, u 1, h][p 1 ], Stab C+ ([p 0 ]) = u 1 c, Stab C ([p 0 ]) = u 1 c h Stab C+ ([p 1 ]) = u 0 c h, Stab C ([p 1 ]) = u 0 c p 1 X How do we get R-matrices : C, C chambers Up to a localization, Stab C is invertible and we set: R C,C = Stab 1 C Stab C, where Stab C : H T (XA ) H T (XA ) Stab C H T (X) H T (XA ) R C,C H T (XA ) ( ) ( ) u h u h 0 Exemple 26 In the basis of fixed points: Stab C+ =, Stab 0 u h C =, ( ) h u u h with u = u 1 u 0, u + h 0 and R C,C + (u) = 1, which is the Yang R-matrix u+h h u More generally, X : Nakajima quiver variety, Q quiver with n vertices give more R- matrices Proof of the Yang-Baxter equation (idea): Stab C R C4,C 3 R C3,C 2 R C2 C 1 = R C4,C 5 R C5,C 6 R C6 C 1 Remark 27 RTT construction: R matrices give quantum groups For Q a quiver, using Nakajima varieties U q (W ), Y Q (Maulik-Okounkov Yangian) is a Hopf algebra 4

3 R matrices and categorification Let g be a simple finite dimensional algebra and n g be a nilpotent sub-algebra Then U q (n) U(g) There is a realization of U q (n) in term of quiver-hecke algebra (Khovanov- Lauda, Rouquier): it may be seen as an analog of affine Hecke algebras of type A: U q (n) K( v N nr v Mod), where Q is a quiver, n is the number of vertices, v N n and v N nr v Mod is a graded finite dimensional module with a convolution product There is a bijection between dual canonical basis and the classes of simple modules (Varagnolo-Vasserot) Moreover: U q (n) has a quantum cluster algebra structure (Bernstein-Zelevinsky) It has a distinguished set of generators (cluster variables) defined inductively by mutation relations (Fomin-Zelevinsky) We have the following: χ χ = Π i χ i + Π j χ j, where χ, χ, χ i, χ j are cluster variables Conjecture 31 Cluster variables belong to the dual canonical basis Notion of categorification of cluster algebras: cluster algebras correspond to certain classes of simple modules Let A be a cluster algebra and M be a monoidal category Then K(M) A (ring isomorphism) Under this isomorphism, { class of simple modules} correspond to {cluster variables} The sequences below correspond to mutation relations 0 j [χ j ] [χ ] [χ] i [χ i ] 0 Exemple 32 Hernandez-Leclerc: using finite dimensional representations of U q (ĝ) Nakajima, Kimura-Qin: using perverse sheaves on quiver varieties Kang-Kashiwara-Kim-Oh: proved the conjecture using KLR algebras Theorem 33 (Kang-Kashiwara-Kim-Oh) v N n R v Mod is a categorification of the cluster algebra U q (n) This theorem proves a conjecture of Fomin-Zelevinsky (see also Qin) Idea of the proof: a crucial point is the construction of normalized R-matrices R norm V W W V (no notion of universal R-matrices) Then we realize mutation relations as exact sequences: 0 Ker V W Rnorm V,W Im 0 V,W : R-matrices give categorification of cluster algebras If Q is a quiver, A Q (quantum) is a cluster algebra generated (as a ring) by cluster variables Let M be a monoidal category such that A Q K 0 (M) (ring isomorphism) cluster variable class of simple modules 5

Exemple 34 (categorical setting) g = sl 3, U q (ŝl 3) For a C, let ev a : U q (ŝl 3) U q (sl 3 ) be the evaluation morphism Let V 1, V 2 be fundamental representations of U q (sl 3 ) For all a C, we have V 1 (a), V 2 (a), 3 dimensional simple U q ( sl ˆ 3 ) representations 0 W V 1 (1) V 1 (q 2 ) Rnorm V 1 (q) V 1 (q 2 ) V 2 (q) 0, where W = ker(r norm ) is simple and V 2 (q) V 1 (q 2 ) V 1 (1) Let C be the category of all finite dimensional representations and C 0 category of all finite dimensional U q ( sl ˆ 3 )-representation V satisfying : C be the [V ] C[[V 1 (1)], [V 1 (q 2 )], [V 2 (q)]], where [V ] is the class of V in K(C) and [V 1 (1)], [V 1 (q 2 )] and [V 2 (q)]] are classes in K(C 0 ) By construction, C 0 is monoidal 1 x z SL 3 N = { 0 1 y }, C[N] = C[x, y, z] 0 0 1 One has C[N] B = {x a y b (xy z) c, a, b, c N} {y a z b (xy z) c, a, b, c N} The set B is the canonical basis We have a ring isomorphism: C[N] ψ K(C 0 ) x [V 1 (1)] y [V 1 (q 2 )] z [V 2 (q)] xy z [V 1 (1) V 1 (q 2 )] [V 2 (q)] = W The map ψ defines a bijection between B and the classes of simple modules Cluster algebra structure Q : 3 1 2 A Q Q(x 1, x 2, x 3 ) The variables x 1, x 2 and x 3 are cluster variables of ψ The mutation relations are X1 = x 1 1 (x 2 + x 3 ), x 1x 1 = x 2 + x 3 One has C A Q C[N] K(C 0 ), x 1 x, x 2 z, x 3 xy z and x 1 y Cluster variables simple modules 4 Quantum integrable system Transform-matrix construction Let V be a finite dimensional representation of U q (ĝ), t V (z) = (Tr V Id)R(z) U q (ĝ)[[z]] = m 0 t V [m]z m ( the t V [m] U q (ĝ) are transformation matrices) By Yang-Baxter equation, t V (z)t W (w) = t W (w)t V (z), the t V [m], for V finite dimensional and m N generate a commutative sub-algebra of U q (ĝ) (apply Tr V W Id) We have actions of K(C) on some spaces: the quantum integrable systems [V ] t V (z) End(W )[[z]] 6

1 "XXZ"-type: W finite dimensional representation of U q (ĝ), action of K(C) on W C[[z]] 2 Quantum KdV-models: W a Fork space over Virasoro algebra Spectrum? Eigenvalues of the operators t V (z) on W? In the case 1: conjecture of Frenkel-Reshetikhin when V, W finite dimensional representations of U q (ĝ) Theorem 41 (Frenkel-Hernandez) The eigenvalues of t V (z) on W can be expressed as a generalized Baxter relation in term of polynomials Exemple 42 g = sl 2, V : 2 dimensional simple module, W tensor product of simple modules Eigenvalues: λ j (Z) = D(z) P j(zq 2 ) P j + A(z) P j(zq 2 ) (z) P j (the Baxter TQ relations) (z) D(z), A(z): universal functions, P j (z) polynomial Remark 43 In general, more than two terms Zeros of the P j (z)? Hint from Langlands duality ODE/IM (integral models) correspondence (Dorey-Tateo) Spectral determinants of Schroedinger operators (differential operators) correspond to eigenvalues of KdV systems Conjectural generalization (Feigin-Frenkel) Affine G opers (without monodromy) are in correspondence with eigen values of quantum KdV systems Masoero-Razmondo-Valeri: on oper side they established Q Q relations, which implies Bethe- Ansatz equations Let C be the finite dimensional representation of U q (ĝ) and O be the category O of representation of U q (ˆb), where ˆb is a Borel sub-algebra (C O) Theorem 44 (Frenkel-Hernandez) The Q Q-system hold in K(O) In particular, the roots of the P j satisfy the Bethe Ansatz equations generically The genericity conditions have been removed by Feigin-Jimbo-Miwa-Mukhin 7