t -deformations of Grothendieck rings as quantum cluster algebras - - PowerPoint PPT Presentation

t-deformations of Grothendieck rings as quantum cluster algebras

Lea Bittmann

Universite Paris-Diderot

June 7, 2018

Lea Bittmann t-deformations of Grothendieck rings

Motivation

Uqpˆ gq : 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 Uqpˆ gq-modules, KpC q its Grothendieck ring. Recall that KpC q “ A rLpˆ λqs , ˆ λ P ˆ P` E . ˆ P` is the set of loop weights, ˆ λ “ ź

iPI,aPCˆ

Yi,a.

Lea Bittmann t-deformations of Grothendieck rings

One has also KpC q “ A rMpˆ λqs , ˆ λ P ˆ P` E , where the prMˆ

λsqˆ λP ˆ P` are called standard modules.

Moreover, rMpˆ λqs “ rLpˆ λqs ` ÿ

µăλ

cˆ

λ,ˆ µrLpˆ

µqs. Nakajima used quiver varieties to compute analogues of Kazhdan-Lusztig polynomials to obtain the coefficients cˆ

λ,ˆ µ (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.

Lea Bittmann t-deformations of Grothendieck rings

Quantum Grothendieck ring

Let t be an indeterminate. The ring KpC q can be t-deformed into pKtpC q, ˚q, a Cptq-algebra with a non-commutative product ˚. For each standard module Mpˆ λq, there is rMpˆ λqst P KtpC q which satisfies rMpˆ λqst

• t“1

“ rMpˆ λqs P KpC q. Define the bar involution : a C-algebra anti-automorphism of KtpC q such that t “ t´1. Proposition (Nakajima) For every simple module Lpˆ λq, there is a unique element rLpˆ λqst of KtpC q satisfying :

1 rLpˆ

λqst “ rLpˆ λqst,

2 rLpˆ

λqst P rMpˆ λqst ` ř

ˆ µăˆ λ t´1Zrt´1srMpˆ

µqst

Lea Bittmann t-deformations of Grothendieck rings

Theorem (Nakajima) For all simple modules Lpˆ λq, rLpˆ λqst

• t“1

“ rLpˆ λqs P KpC q. Moreover, if we write rMpˆ λqst “ rLpˆ λqst ` ÿ

ˆ µăˆ λ

t´1Zˆ

µ,ˆ λpt´1qrLpˆ

µqst, then Zˆ

µ,ˆ λptq P Nrts, and

Zˆ

µ,ˆ λp1q “ rMpˆ

λq, Lpˆ µqs

Lea Bittmann t-deformations of Grothendieck rings

Category O

Let Uqpˆ bq be the Borel subalgebra of Uqpgq (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 L˘

i,a (i P I, a P Cˆ), simple

infinite dimensional representations of highest ℓ-weights Ψ˘

i,a,

such that Yi,a “ rωisΨi,aq´1 Ψi,aq . ù For g “ sl2, 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.

Lea Bittmann t-deformations of Grothendieck rings

Some subcategories

Let Γ be one of the two connected components of the quiver with vertices I ˆ Z and arrows pi, rq Ñ pj, sq iff s “ r ` Ci,j. Let V be its set of vertices. Example : g “ sl4: . . . p2, 1q

• .

. . p1, 0q

• p3, 0q
• p2, ´1q
• p1, ´2q
• p3, ´2q
• p2, ´3q
• p1, ´4q
• .

. . p3, ´4q

• Lea Bittmann

t-deformations of Grothendieck rings

Category CZ

Let ˆ P`,Z be the ℓ-weights of the form: ź

pi,rqPV

Y ui,r

i,qr`1

CZ: full subcategory of C of representations whose composition factors are of the form Lpˆ λq, for ˆ λ P ˆ P`,Z. The non-commutative Cptq-algebra KtpCZq belongs to the quantum torus pY , ˚q, which is generated by the pY ˘

i,qr`1qpi,rqPV , and such that,

Yi,qr ˚ Yj,qs “ tNi,jps´rqYj,qs ˚ Yi,qr .

Lea Bittmann t-deformations of Grothendieck rings

Category O`

Z

O`

Z : full subcategory of O of representations whose

composition factors are of the form Lpˆ λq, such that ˆ λ “ ź

pi,rqPV

Ψui,r

i,qr

ź

pj,sqPV

Y vj,s

j,qs`1.

Theorem (Hernandez-Leclerc, 2016) KpO`

Z q

– ApΓqˆ bEℓ ” L`

i,r

ı ÞÑ zi,r . ù Idea: Built KtpO`

Z q as a Quantum cluster algebra.

Lea Bittmann t-deformations of Grothendieck rings

Extended quantum torus

Recall that Yi,qr`1 “ rωis Ψi,qr Ψi,qr`2 “ rωis ˜ Yi,qr`1. Proposition (B.) There exists a quantum torus pT , ˚q, generated by the pΨ˘

i,qr qpi,rqPV , such that

˜ Y Ă T . The pΨi,qr q satisfy Ψi,qr ˚ Ψj,qs “ tΛi,jps´rqΨj,qs ˚ Ψi,qr , Let Λ : ppi, rq, pj, sqq ÞÑ Λi,jps ´ rq.

Lea Bittmann t-deformations of Grothendieck rings

Quantum cluster algebra

Quantum cluster algebras: non commutative t-deformations

• f 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 KtpO`

Z q :“ ApΓ, Λqˆ

bEℓ.

Lea Bittmann t-deformations of Grothendieck rings

Example : g “ sl2

In this case, the complete classification of the simple and prime simple representations is known. We can define pq, tq-characters rLst P KtpO`

Z q for all simple

representations. If L is finite-dimensional, its pq, tq-characters rLst is the same as the one in KtpCZq. The mutation relations provide some insightful relations in KtpO`

Z q.

Lea Bittmann t-deformations of Grothendieck rings

Fundamental example : categorified Baxter TQ relations Let Vq´1 be the two-dimensional evaluation representation of Uqpˆ sl2q, of highest ℓ-weight Yq´1. We have the following relation in KtpO`

Z q:

rVq´1st ˚ rL`

1 st “ t´ 1

2 rL`

q´2st ` t

1 2 rL`

q2st

ù Linked to Baxter’s TQ relations for the eigenvalues of the transfer matrix of the corresponding quantum integrable system (XXZ spin chain model).

Lea Bittmann t-deformations of Grothendieck rings

Next ?

Define some pq, tq-characters rLst for all simple modules for all types. Prove that KtpCZq Ă KtpO`

Z q.

What are the standard modules ?

Lea Bittmann t-deformations of Grothendieck rings

Thank you!

Lea Bittmann t-deformations of Grothendieck rings