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 thetheory 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 ofaffine opers.

Contents
1 Algebraic solution (Drinfeld)

12 Geometric construction (theory stable envelop, Maulik-Okounkov

3

3 R matrices and categorification

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4 Quantum integrable system

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1Algebraic solution (Drinfeld)

Définition 1.1. Let A be an algebra over C and z be an indeterminate. The Yang-Baxter
equation (with spectral parameters) is as follows:
R12 (z)R13 (zw)R23 (w) = R23 (w)R13 (zw)R12 (z),where R(z) ∈ (A⊗A)(z), R12 (z) = R(z)⊗1, R23 (z) = 1⊗R(z) and R13 (z) = (P ⊗Id)R23 (z);P is a "twist".
Exemple 1.2. A fundamental (historical) example (Yang R-matrix): A = End(C2 ), q ∈ C∗ ,
1000
0 q−1 (z−1) 1−q−2 0z−q −2z−q −2R(z) =  z(1−q−2 )(z−1)q −10 z−q−20
z−q −100011.
It is attached to the 6-vertex model (ice) of XXZ-spin chain model.
hIf one writes z = eu , q = e 2 , then!
q −1 (z−1)
1−q −2
1u h−2−2z−qz−q∼.
(z−1)q −1z(1−q −2h uh,u→0 u + h−2−1z−qz−qAlgebraic construction: V is a 2 dimensional representation of Uq (slˆ2 ) (affine quantumgroup).
We can generalize by replacing sl2 by g a general simple finite dimensional Lie algebraand V by a finite dimensional representation of Uq (ĝ). This gives rise to many R-matrices.
By Drinfeld-Jimbo, we have the following:quantizationg/ Uq (g)/4 Uq (ĝ)

affinization
quantization*affine Kac − Moody ĝ = Cc ⊕ C[t±1 ] ⊗ g
Theorem 1.3. (Drinfeld)"Commutative diagram" gives 2 presentation of Uq (ĝ).
Remark 1.4. Uq (ĝ) is a Hopf algebra.
ˆ q (ĝ)[[z]], where
Theorem 1.5. Uq (ĝ) has a universal non trivial R-matrix: R(z) ∈ (Uq (ĝ)⊗Uˆ⊗ is some completion of the tensor product for some filtration, such that from two finite
dimensional representations V, W of Uq (ĝ), we get P RV,W (z) : V ⊗ W [[z]] → W ⊗ V [[z]],where RV,W (z) is the image of R(z) in End(V ⊗ W )[[z]]: we get a well defined morphism ofUq (ĝ)-modules.
Proposition 1.6. If V, W are simple and finite dimensional, there exists fV,W ∈ C((z)) suchthat fV,W (z)RV,W (z) is rational in End((V ⊗ W )) ⊗ C(z). Let d ∈ Z be the order of 1 as apole of RV,W (z)fV,W (z). Then limz→1 [(z − 1)d P fV,W (z)RV,W (z)] := Rnorm
V,W : V ⊗ W → W ⊗ Vis a non-zero Uq (ĝ)-module morphism.
Remark 1.7. In general we do not know the orders of poles and the zeros of rational points.
Exemple 1.8. see example above, g = sl2 , V = V (1), W = W (q 2 ), evaluation representationof Uq (slˆ2 ) (morphism of algebra Uq (slˆ2 ) → Uq (slˆ2 )). Then d = 1, i.e 1 is a pole of order 1 andRnorm
V,W has rank 1 (not invertible).
Remark 1.9. If U, V, W are finite dimensional representations of Uq (ĝ):
Id⊗P RU,W (zw)

P RU,V (f )⊗Id/V ⊗W ⊗UV ⊗6 U ⊗W

(U ⊗V ⊗W

W 6⊗ V ⊗ UId⊗P RV,W (w)(/W ⊗U ⊗VU ⊗W ⊗V

By the Yang-Baxter equation, this diagram is commutative.

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Geometric construction (theory stable envelop, MaulikOkounkov

We consider the couple (X, ω), where X is a quasi-projective, non singular algebraic varietyand ω is a holomorphic simplectic form. We have an action of (C∗ )R+n ' T ⊃ A ' (C∗ )R .
Hypothesis:• Cω ⊂ H 0 (Ω2 ω) is stable for the induced action of T . We have a corresponding characterh : T → C∗ and A ⊂ Kerh
• there exists a proper T -equivariant affine map F : X → X0 satisfying an additional
technical formality condition.
Exemple 2.1.
• A = (C∗ )n+1 , T = A × C∗ , X = T ∗ Pn (action on coordinates of Pn ).
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 . It has a structure of a direct sum of 1 dimensional representations of A.
ADéfinition 2.2. Let ∆ be the set of characters of A corresponding to A-simple modules
occurring in N (X A ).
One has ∆ ⊂ XR∗ = C(A) ⊗Z R, where C(A) is the group of characters of A. One has
XR ' Rn ⊂ Lie(A).
⊥For α ∈ ∆, we have the hyperplane{v F∈ XR |α(v) = 0}.
S α =We have the decomposition XR \ α∈∆ α⊥ = i Ci , where the Ci are the open chambers.
For a connected component Z ⊂ X A and a chamber C, we set
leaf C (Z) = {x ∈ X| lim (σ(z).x) exists and is in Z}
z→0

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 0 FC Z if leaf C Z ∩ Z 0 6= ∅. The slope of a connected component Z is thenSlopeC (Z) = Z 0 Z leaf C (Z 0 ).
Exemple 2.3. X = T ∗ P1 , ∆ = {α, −α}, α(a, b) = au0 b−u1 where u0 , u1 are characters forthe action on A.
One has XR = R2 = C+ t α⊥ t C− , where C = {(x, y) ∈ R2 |u0 x > u1 y} for  ∈ {−, +}.
Let p0 = [1 : 0], p1 = [0 : 1]. One has P1 ⊂ T ∗ P1 .
One has leaf C+ ({p0 }) = Tp∗0 (P1 ), leaf C− ({p1 }) = Tp∗1 (P1 ), leaf C+ ({p1 }) = P1 \{p1 }, leaf C+ ({p0 }) =P1 \{p0 }.
Let HT• (X), HT• (X A ) be the T -equivariant cohomology.
Theorem 2.4. (Maulik-Okounkov) There exists a unique StabC : HT• (X A ) → HT• (X) morphism of HT• ({pt})-modules such that forγ ∈ HT• (Z), with Z a connected component of X A
and µ = StabC (γ), we have :1. Supp(µ) ⊂ SlopeC (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 0 C Z, degA (µ|Z 0 ) < codimZ (Z 0 ).
Idea of construction: based on the construction of a Lagrangian correspondence such that
p1 is proper.
L ⊂ X × XA
p1

XA

p2x&X

∗Then StabC = (p1 )∗ ◦ (p2 ) .
Exemple 2.5. X = T ∗ (P1 ), u0 , u1 , h characters, HT• ({pt}) = C[u0 , u1 , h], HT• (X) = C[u0 , u1 , h]⊕C[u0 , u1 , h]c, where c is the first Cherr character of O(−1).
HT• (X A ) = C[u0 , u1 , h][p0 ] ⊕ C[u0 , u1 , h][p1 ],StabC+ ([p0 ]) = u1 − c, StabC− ([p0 ]) = u1 − c − h
StabC+ ([p1 ]) = u0 − c − h, StabC− ([p1 ]) = u0 − c.
How do we get R-matrices : C, C 0 chambers. Up to a localization, StabC 0 is invertible•A•A
and we set: RC 0 ,C = Stab−1
C 0 ◦ StabC , where StabC : HT (X ) → HT (X ).
HT• (X)
StabCe9

StabC 0HT• (X A )RC,C 0/ H • (X A )T−u −h−u − h 0Exemple 2.6. In the basis of fixed points: StabC+ =, StabC− =,−hu 0 u − hu h1with u = u1 − u0 , u + h 6= 0 and RC− ,C+ (u) = u+h, which is the Yang R-matrix.
h uMore generally, X : Nakajima quiver variety, Q quiver with n vertices give more Rmatrices.
Proof of the Yang-Baxter equation (idea):

RC4 ,C3 RC3 ,C2 RC2 C1 = RC4 ,C5 RC5 ,C6 RC6 C1 .
Remark 2.7. RTT construction: R matrices give quantum groups.
For Q a quiver, using Nakajima varieties Uq (W ), YQ (Maulik-Okounkov Yangian) is a
Hopf algebra.
43R matrices and categorification

Let g be a simple finite dimensional algebra and n ⊂ g be a nilpotent sub-algebra. ThenUq (n) ⊂ U(g). There is a realization of Uq (n) in term of quiver-Hecke algebra (KhovanovLauda, Rouquier): it may be seen as an analog of affine Hecke algebras of type A: Uq (n) 'K(⊕v∈Nn Rv − Mod), where Q is a quiver, n is the number of vertices, v ∈ Nn and ⊕v∈Nn Rv −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: Uq (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 = Πi χi + Πj χj , where χ, χ∗, χi , χj are cluster variables.
Conjecture 3.1. Cluster variables belong to the dual canonical basis.
Notion of categorification of cluster algebras: cluster algebras correspond to certainclasses 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
OO0→[χj ] → [χ∗ ] ⊗ [χ] →[χi ] → 0ji.
Exemple 3.2.

• Hernandez-Leclerc: using finite dimensional representations of Uq (ĝ)• Nakajima, Kimura-Qin: using perverse sheaves on quiver varieties• Kang-Kashiwara-Kim-Oh: proved the conjecture using KLR algebras.
Theorem
3.3. (Kang-Kashiwara-Kim-Oh)
LRv∈Nn v − M od is a categorification of the cluster algebra Uq (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 Rnorm
V,W :V ◦ W → W ◦ V (no notion of universal R-matrices).
Then we realize mutation relations as exact sequences:
0 → Ker → V ◦ WRnorm
V,W→ Im → 0.

R-matrices give categorification of cluster algebras.
If Q is a quiver, AQ (quantum) is a cluster algebra generated (as a ring) by cluster
variables.
Let M be a monoidal category such that AQ ' K0 (M) (ring isomorphism).
cluster variable ⊂ class of simple modules.

5ˆ 3 ). For a ∈ C∗ , let eva : Uq (slˆ 3) →Exemple 3.4.
• (categorical setting) g = sl3 , Uq (slUq (sl3 ) be the evaluation morphism.
Let V1 , V2 be fundamental representations of Uq (sl3 ). For all a ∈ C∗ , we have V1 (a),V2 (a), 3 dimensional simple Uq (slˆ3 ) representations.
0 ⊂ W ⊂ V1 (1) ⊗ V1 (q 2 )RnormV1 (q)⊗V1 (q 2 )→V2 (q) → 0,

where W = ker(Rnorm ) is simple and V2 (q) ⊂ V1 (q 2 ) ⊗ V1 (1).
Let C be the category of all finite dimensional representations and C0 ⊂ C be thecategory of all finite dimensional Uq (slˆ3 )-representation V satisfying :[V ] ∈ C[[V1 (1)], [V1 (q 2 )], [V2 (q)]],where [V ] is the class of V in K(C) and [V1 (1)], [V1 (q 2 )] and [V2 (q)]] are classes in K(C0 ).
By construction, C0 is monoidal.
1 x z• SL3 ⊃ N = {0 1 y }, C[N ] = C[x, y, z].
0 0 1
One has C[N ] ⊃ B = {xa 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.
ψC[N ] ' K(C0 )x ↔ [V1 (1)]We have a ring isomorphism:.
y ↔ [V1 (q 2 )]z ↔ [V2 (q)]xy − z ↔ [V1 (1) ⊗ V1 (q 2 )] − [V2 (q)] = W
The map ψ defines a bijection between B and the classes of simple modules.
Cluster algebra structure Q : 3 → 1 → 2AQ ⊂ Q(x1 , x2 , x3 ). The variables x1 , x2 and x3 are cluster variables of ψ. The mutation∗
relations are X1∗ = x−1
1 (x2 + x3 ), x1 x1 = x2 + x3 .
One has C ⊗ AQ ' C[N ] ' K(C0 ), x1 ↔ x, x2 ↔ z, x3 ↔ xy − z and x∗1 ↔ y.
Cluster variables ⊂ simple modules.

4Quantum integrable system

Transform-matrix construction Let
dimensional representation of Uq (ĝ),
P V be a finite
mtV (z) = (TrV ⊗ Id)R(z) ∈ Uq (ĝ)[[z]] = m≥0 tV [m]z ( the tV [m] ∈ Uq (ĝ) are transformationmatrices).
By Yang-Baxter equation, tV (z)tW (w) = tW (w)tV (z), the tV [m], for V finite dimensionaland m ∈ N generate a commutative sub-algebra of Uq (ĝ) (apply TrV ⊗W ⊗ Id).
We have actions of K(C) on some spaces: the quantum integrable systems.
[V ] → tV (z) → End(W )[[z]].
61. "XXZ"-type: W finite dimensional representation of Uq (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 tV (z) on W ? In the case 1: conjecture ofFrenkel-Reshetikhin when V, W finite dimensional representations of Uq (ĝ)
Theorem 4.1. (Frenkel-Hernandez)
The eigenvalues of tV (z) on W can be expressed as a generalized Baxter relation in termof polynomials.
Exemple 4.2. g = sl2 , V : 2 dimensional simple module, W tensor product of simple modP (zq 2 )P (zq −2 )ules. Eigenvalues: λj (Z) = D(z) jPj (z) + A(z) Pj j (z) (the Baxter TQ relations)D(z), A(z): universal functions, Pj (z) polynomial.
Remark 4.3. In general, more than two terms.
Zeros of the Pj (z) ? Hint from Langlands duality ODE/IM (integral models) correspondence (Dorey-Tateo).
Spectral determinants of Schroedinger operators (differential operators) correspond toeigenvalues 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 QQ̃ relations, which implies
Bethe- Ansatz equations.
ˆ and O be the category O ofLet C be the finite dimensional representation of Uq (g)representation of Uq (b̂), where b̂ is a Borel sub-algebra (C ⊂ O)
Theorem 4.4. (Frenkel-Hernandez) The QQ̃-system hold in K(O).
In particular, the roots of the Pj satisfy the Bethe Ansatz equations generically. The
genericity conditions have been removed by Feigin-Jimbo-Miwa-Mukhin.

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