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 U q p ˆ g q : 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 p ˆ g q -modules, K p C q its Grothendieck ring. Recall that A E r L p ˆ λ qs , ˆ λ P ˆ K p C q “ P ` . ˆ P ` is the set of loop weights , ˆ ź λ “ Y i , a . i P I , a P C ˆ Lea Bittmann t -deformations of Grothendieck rings

One has also A E r M p ˆ λ qs , ˆ λ P ˆ K p C q “ P ` , where the pr M ˆ λ sq ˆ P ` are called standard modules . λ P ˆ Moreover, r M p ˆ λ qs “ r L p ˆ ÿ λ qs ` c ˆ µ r L p ˆ µ qs . λ, ˆ µ ă λ Nakajima used quiver varieties to compute analogues of Kazhdan-Lusztig polynomials to obtain the coefficients c ˆ µ (for λ, ˆ 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 K p C q can be t -deformed into p K t p C q , ˚q , a C p t q -algebra with a non-commutative product ˚ . For each standard module M p ˆ λ q , there is r M p ˆ λ qs t P K t p C q which satisfies � r M p ˆ “ r M p ˆ λ qs t � λ qs P K p C q . � � t “ 1 Define the bar involution : a C -algebra anti-automorphism of K t p C q such that t “ t ´ 1 . Proposition (Nakajima) For every simple module L p ˆ λ q , there is a unique element r L p ˆ λ qs t of K t p C q satisfying : 1 r L p ˆ λ qs t “ r L p ˆ λ qs t , 2 r L p ˆ λ qs t P r M p ˆ λ t ´ 1 Z r t ´ 1 sr M p ˆ λ qs t ` ř µ qs t µ ă ˆ ˆ Lea Bittmann t -deformations of Grothendieck rings

Theorem (Nakajima) For all simple modules L p ˆ λ q , � r L p ˆ “ r L p ˆ λ qs t � λ qs P K p C q . � � t “ 1 Moreover, if we write r M p ˆ λ qs t “ r L p ˆ ÿ t ´ 1 Z ˆ λ p t ´ 1 qr L p ˆ λ qs t ` µ qs t , µ, ˆ µ ă ˆ ˆ λ then Z ˆ λ p t q P N r t s , and µ, ˆ λ p 1 q “ r M p ˆ Z ˆ λ q , L p ˆ µ qs µ, ˆ Lea Bittmann t -deformations of Grothendieck rings

Category O Let U q p ˆ b q be the Borel subalgebra of U q p g q (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 C ˆ ), simple infinite dimensional representations of highest ℓ -weights Ψ ˘ i , a , such that Y i , a “ r ω 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. 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 p i , r q Ñ p j , s q iff s “ r ` C i , j . Let V be its set of vertices. Example : g “ sl 4 : . . . . . p 2 , 1 q . p 1 , 0 q p 3 , 0 q p 2 , ´ 1 q p 1 , ´ 2 q p 3 , ´ 2 q p 2 , ´ 3 q . . p 1 , ´ 4 q . p 3 , ´ 4 q Lea Bittmann t -deformations of Grothendieck rings

Category C Z Let ˆ P ` , Z be the ℓ -weights of the form: ź Y u i , r i , q r ` 1 p i , r qP V C Z : full subcategory of C of representations whose composition factors are of the form L p ˆ λ q , for ˆ λ P ˆ P ` , Z . The non-commutative C p t q -algebra K t p C Z q belongs to the quantum torus p Y , ˚q , which is generated by the p Y ˘ i , q r ` 1 q p i , r qP V , and such that, Y i , q r ˚ Y j , q s “ t N i , j p s ´ r q Y j , q s ˚ Y i , q r . 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 L p ˆ λ q , such that Ψ u i , r Y v j , s ˆ ź ź λ “ j , q s ` 1 . i , q r p i , r qP V p j , s qP V Theorem (Hernandez-Leclerc, 2016) K p O ` A p Γ q ˆ Z q – b E ℓ . ” ı L ` ÞÑ z i , r i , r ù Idea: Built K t p O ` Z q as a Quantum cluster algebra . Lea Bittmann t -deformations of Grothendieck rings

Extended quantum torus Recall that Y i , q r ` 1 “ r ω i s Ψ i , q r Ψ i , q r ` 2 “ r ω i s ˜ Y i , q r ` 1 . Proposition (B.) There exists a quantum torus p T , ˚q , generated by the p Ψ ˘ i , q r q p i , r qP V , such that ˜ Y Ă T . The p Ψ i , q r q satisfy Ψ i , q r ˚ Ψ j , q s “ t Λ i , j p s ´ r q Ψ j , q s ˚ Ψ i , q r , Let Λ : pp i , r q , p j , s qq ÞÑ Λ i , j p s ´ r q . Lea Bittmann t -deformations of Grothendieck rings

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 p Λ , B q is a compatible pair. Then, Definition K t p O ` Z q : “ A p Γ , Λ q ˆ b E ℓ . Lea Bittmann t -deformations of Grothendieck rings

Example : g “ sl 2 In this case, the complete classification of the simple and prime simple representations is known. We can define p q , t q -characters r L s t P K t p O ` Z q for all simple representations. If L is finite-dimensional, its p q , t q -characters r L s t is the same as the one in K t p C Z q . The mutation relations provide some insightful relations in K t p O ` Z q . Lea Bittmann t -deformations of Grothendieck rings

Fundamental example : categorified Baxter TQ relations Let V q ´ 1 be the two-dimensional evaluation representation of U q p ˆ sl 2 q , of highest ℓ -weight Y q ´ 1 . We have the following relation in K t p O ` Z q : 1 s t “ t ´ 1 1 r V q ´ 1 s t ˚ r L ` 2 r L ` 2 r L ` q ´ 2 s t ` t 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). Lea Bittmann t -deformations of Grothendieck rings

Next ? Define some p q , t q -characters r L s t for all simple modules for all types. Prove that K t p C Z q Ă K t p O ` Z q . What are the standard modules ? Lea Bittmann t -deformations of Grothendieck rings

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