t-deformations of Grothendieck rings as quantum cluster algebras

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1 as quantum cluster algebras Universite Paris-Diderot June 7, 2018

2 Motivation U q pĝq : untwisted quantum Kac-Moody affine algebra of simply laced type, where q P C is not a root of unity, C : the category of finite-dimensional U q pĝq-modules, KpC q its Grothendieck ring. Recall that A KpC q rlpˆλqs, ˆλ P ˆP` ˆP` is the set of loop weights, ˆλ ź ipi,apcˆ Y i,a. E.

3 One has also A KpC q rmpˆλqs, ˆλ P ˆP` E, where the prmˆλsqˆλp are called standard modules. ˆP` Moreover, rmpˆλqs rlpˆλqs ` ÿ cˆλ,ˆµ rlpˆµqs. µăλ Nakajima used quiver varieties to compute analogues of Kazhdan-Lusztig polynomials to obtain the coefficients cˆλ,ˆµ (for the ADE case). For the standard modules, dimensions of the eigenspaces and characters are known, we want the same information on the simple modules.

4 Quantum Grothendieck ring Let t be an indeterminate. The ring KpC q can be t-deformed into pk t pc q, q, a Cptq-algebra with a non-commutative product. For each standard module Mpˆλq, there is rmpˆλqs t P K t pc q which satisfies rmpˆλqs t t 1 rmpˆλqs P KpC q. Define the bar involution K t pc q such that t t 1. : a C-algebra anti-automorphism of Proposition (Nakajima) For every simple module Lpˆλq, there is a unique element rlpˆλqs t of K t pc q satisfying : 1 rlpˆλqs t rlpˆλqs t, 2 rlpˆλqs t P rmpˆλqs t ` řˆµăˆλ t 1 Zrt 1 srmpˆµqs t

5 Theorem (Nakajima) For all simple modules Lpˆλq, rlpˆλqs t t 1 rlpˆλqs P KpC q. Moreover, if we write rmpˆλqs t rlpˆλqs t ` ÿ t 1 Zˆµ,ˆλ pt 1 qrlpˆµqs t, then Zˆµ,ˆλptq P Nrts, and ˆµăˆλ Zˆµ,ˆλp1q rmpˆλq, Lpˆµqs

6 Category O Let U q pˆbq be the Borel subalgebra of U q pgq (in the sense of Drinfeld-Jimbo presentation). Hernandez-Jimbo : category O of representations for this algebra. It contains: the finite-dimensional representations C, the prefundamental representations Lĭ,a (i P I, a P Cˆ), simple infinite dimensional representations of highest l-weights Ψĭ,a, such that Y i,a rω i s Ψ i,aq 1 Ψ i,aq. ù For g sl 2, these appeared naturally in the works of Bazhanov-Lukyanov-Zamolodchikov, under the name q-oscillator representations. They are linked to the eigenvalues of transfer matrices of quantum integrable systems.

7 Some subcategories Let Γ be one of the two connected components of the quiver with vertices I ˆ Z and arrows Let V be its set of vertices. Example : g sl 4 : pi, rq Ñ pj, sq iff s r ` C i,j.. p2, 1q p1, 0q p1, 2q p1, 4q p2, 1q p2, 3q. p3, 0q p3, 2q. p3, 4q

8 Category C Z Let ˆP`,Z be the l-weights of the form: ź pi,rqpv Y u i,r i,q r`1 C Z : full subcategory of C of representations whose composition factors are of the form Lpˆλq, for ˆλ P ˆP`,Z. The non-commutative Cptq-algebra K t pc Z q belongs to the quantum torus py, q, which is generated by the py i,q r`1 q pi,rqpv, and such that, Y i,q r Y j,q s t N i,jps rq Y j,q s Y i,q r.

9 Category O`Z O`Z : full subcategory of O of representations whose composition factors are of the form Lpˆλq, such that ˆλ ź pi,rqpv Ψ u i,r i,q r ź pj,sqpv Y v j,s j,q s`1. Theorem (Hernandez-Leclerc, 2016) KpO`Z q ApΓq ˆbE l Lì,r ı ÞÑ z i,r. ù Idea: Built K t po`z q as a Quantum cluster algebra.

10 Extended quantum torus Recall that Proposition (B.) Y i,q r`1 rω i s Ψ i,q r Ψ i,q r`2 rω i sỹ i,q r`1. There exists a quantum torus pt, q, generated by the pψĭ,qr q pi,rqpv, such that Y Ă T. The pψ i,q r q satisfy Ψ i,q r Ψ j,q s t Λ i,jps rq Ψ j,q s Ψ i,q r, Let Λ : ppi, rq, pj, sqq ÞÑ Λ i,j ps rq.

11 Quantum cluster algebra Quantum cluster algebras: non commutative t-deformations of cluster algebras. Lives inside a quantum torus, such that the mutations relations (i.e. the exchange matrix) are compatible with the t-commutation relations. Proposition Let B be the exchange matrix associated to the infinite quiver Γ. Then pλ, Bq is a compatible pair. Then, Definition K t po`z q : ApΓ, Λq ˆbE l.

12 Example : g sl 2 In this case, the complete classification of the simple and prime simple representations is known. We can define pq, tq-characters rls t P K t po`z q for all simple representations. If L is finite-dimensional, its pq, tq-characters rls t is the same as the one in K t pc Z q. The mutation relations provide some insightful relations in K t po`z q.

13 Fundamental example : categorified Baxter TQ relations Let V q 1 be the two-dimensional evaluation representation of U q pŝl 2 q, of highest l-weight Y q 1. We have the following relation in K t po`z q: rv q 1s t rl`1 s t t 1 2 rl` q 2 s t ` t 1 2 rl` q 2 s t ù Linked to Baxter s TQ relations for the eigenvalues of the transfer matrix of the corresponding quantum integrable system (XXZ spin chain model).

14 Next? Define some pq, tq-characters rls t for all simple modules for all types. Prove that K t pc Z q Ă K t po`z q. What are the standard modules?

15 Thank you!