Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

Zhang, Huafeng

doi:10.1007/s00220-017-3022-7

Cited by 6 publications

(12 citation statements)

References 55 publications

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“…k+t+1,0 → 0. These exact sequences hold for affine quantum (super)groups [22,54]. In the super case the proof is more delicate since Lemma 3.2 (3) fails.…”

Section: Kirillov-reshetikhin Modulesmentioning

confidence: 94%

Elliptic quantum groups and Baxter relations

Zhang

2018

Alg. Number Th.

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We introduce a category O of modules over the elliptic quantum group of sl N with well-behaved q-character theory. We construct asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov-Reshetikhin modules. In the Grothendieck ring of this category we prove two types of identities: generalized Baxter relations in the spirit of Frenkel-Hernandez between finite-dimensional modules and asymptotic modules; three-term Baxter TQ relations of infinite-dimensional modules.1 2 HUAFENG ZHANG Such R-matrices R(z; λ) appeared previously in face-type integrable models [20,38]; for instance, the R-matrix of the Andrews-Baxter-Forrester model comes from two-dimensional irreducible modules of E τ, (sl 2 ), as does the 6-vertex model from the affine quantum group U (Lsl 2 ). The definition of E τ, (sl N ) in [17], by RLL exchange relations, is in the spirit of Faddeev-Reshetikhin-Takhatajan, originated from Quantum Inverse Scattering Method. We mention that elliptic R-matrices describe the monodromy of the quantized Knizhnik-Zamolodchikov equation associated with representations of affine quantum groups, e.g. [28,29,43,50].Recently Aganagic-Okounkov [1] proposed the elliptic stable envelope in equivariant elliptic cohomology, as a geometric framework to obtain elliptic R-matrices. This was made explicit [18] for cotangent bundles of Grassmannians, resulting in tensor products of two-dimensional irreducible representations of E τ, (sl 2 ). The higher rank case of sl N was studied later by H. Konno [46].Meanwhile, Nekrasov-Pestun-Shatashvili [49] from the 6d quiver gauge theory predicted the elliptic quantum group associated to an arbitrary Kac-Moody algebra, the precise definition of which (as an associative algebra) was proposed by . See also [52] in the context of quiver geometry.We are interested in the representation theory of E τ, (g) with ∈ C generic. The formal twist constructions [11,40] from U (Lg) might reduce the problem to the representation theory of affine quantum groups, which is a subject developed intensively in the last three decades from algebraic, geometric and combinatorial aspects. However loc.cit. involve formal power series of and infinite products in the comultiplication of E τ, (g). Some of these divergence issues was addressed [12] by Etingof-Moura, who defined a fully faithful tenor functor between representation categories of BGG type for U (Lsl N ) and E τ, (sl N ). Towards this functor not much is known: its image, the induced homomorphism of Grothendieck rings, etc.In this paper we study representations of E τ, (sl N ) via the RLL presentation [17] so as to bypass affine quantum groups, yet along the way we borrow ideas from the affine case. Compared to other works [7,12,19,32,44,45,51,52], our approach emphasizes more on the Grothendieck ring structure of representation category. It is a higher rank extension of a recent joint work with G. Felder [21].The presence of the dynamical parameter λ is one of the technical difficulties of elliptic quantum groups. ...

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“…k+t+1,0 → 0. These exact sequences hold for affine quantum (super)groups [22,54]. In the super case the proof is more delicate since Lemma 3.2 (3) fails.…”

Section: Kirillov-reshetikhin Modulesmentioning

confidence: 94%

Elliptic quantum groups and Baxter relations

Zhang

2018

Alg. Number Th.

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“…In the sequels [13,36,37] we define Baxter Q-operators from transfer matrices of the ρ c , and interpret Theorem 6.11 as generalized T-Q relations of transfer matrices. Surprisingly the spin parameter c becomes the spectral parameter of Q-operators.…”

Section: H Zhangmentioning

confidence: 99%

“…Then e − i F k,l = F k,l e − i as in (1), using ∆(e − i ). For i = r, we adapt the proof of [36,Lemma 7.6]. We fix l = m + 1 in (2).…”

Section: H Zhangmentioning

confidence: 99%

“…In both W (r) a;c and L + r,a , ω ∞ is the unique (up to 7 In [36], Drinfeld second realization arising from a different Gauss decomposition from Section 6 is used to resolve the issue in footnote 2. This results in different parameterizations of highest -weights.…”

Section: Consider the Action Of The S II On Wmentioning

confidence: 99%

“…Such formula appeared earlier in the context of transfer matrices of quantum integrable systems attached to U q ( g) [30]. In a recent preprint [36], the tableau-sum q-character formula has been extended to ev * a (L * (λ)), (ev a ) * L(λ), (ev a ) * (L * (λ)),…”

Section: Rationality Of -Weights Of F Inite-dimensional Modulesmentioning

confidence: 99%

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Asymptotic Representations of Quantum Affine Superalgebras

Zhang¹

2017

SIGMA

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Abstract. We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called q-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of Frenkel-Hernandez for representations of the full quantum group.

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Theta Series for Quantum Loop Algebras and Yangians

Zhang

2024

Commun. Math. Phys.

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Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

Cited by 6 publications

References 55 publications

Elliptic quantum groups and Baxter relations

Elliptic quantum groups and Baxter relations

Asymptotic Representations of Quantum Affine Superalgebras

Theta Series for Quantum Loop Algebras and Yangians

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