Baxter operators and asymptotic representations

Felder, Giovanni; Zhang, Huafeng

doi:10.1007/s00029-017-0320-z

Cited by 7 publications

(28 citation statements)

References 41 publications

(59 reference statements)

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“…Since the elliptic quantum groups will act on M-vector spaces via difference operators, which are in general not M-linear, we need to recall some basis constructions about difference operators. Our exposition follows largely [14], with minor modifications as in [21].…”

Section: Elliptic Quantum Groups and Their Representationsmentioning

confidence: 99%

“…Let us verify Property (M3) of Definition 1.4. The idea is almost the same as that of [21,Prop.3.9], which in turn followed…”

Section: 3mentioning

confidence: 99%

“…By "uniform" we mean the multipliers with respect to z + 1 and z + τ only depend on (a, z, ℓ). This is because the transfer matrix construction in [21] is based on a slightly different elliptic quantum group; see Footnote 9.…”

Section: Transfer Matrices and Baxter Operatorsmentioning

confidence: 99%

“…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. To resolve this, we need a commuting family of elliptic Cartan currents φ j (z) ∈ E τ, (sl N ) for j ∈ J := {1, 2, · · · , N − 1}.…”

mentioning

confidence: 96%

“…They act as difference operators on an E τ, (sl N )-module V , and their matrix entries are meromorphic functions of (z, λ) ∈ C × h where h denotes the Cartan subalgebra of sl N . As in [21], we impose the following triangularity condition: 1 (i) there exists a basis of V , with respect to which the matrices φ j (z) are upper triangular and their diagonal entries are independent of λ.Our category O is the full subcategory of category BGG [12] of E τ, (sl N )-modules subject to Condition (i); see Definition 1.7. It is abelian and monoidal.…”

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confidence: 99%

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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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Section: Elliptic Quantum Groups and Their Representationsmentioning

confidence: 99%

“…Let us verify Property (M3) of Definition 1.4. The idea is almost the same as that of [21,Prop.3.9], which in turn followed…”

Section: 3mentioning

confidence: 99%

Section: Transfer Matrices and Baxter Operatorsmentioning

confidence: 99%

mentioning

confidence: 96%

mentioning

confidence: 99%

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Elliptic quantum groups and Baxter relations

Zhang

2018

Alg. Number Th.

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Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

Zhang

2017

Commun. Math. Phys.

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Associated to quantum affine general linear Lie superalgebras are two families of short exact sequences of representations whose first and third terms are irreducible: the Baxter TQ relations involving infinite-dimensional representations; the extended T-systems of Kirillov-Reshetikhin modules. We make use of these representations over the full quantum affine superalgebra to define Baxter operators as transfer matrices for the quantum integrable model and to deduce Bethe Ansatz Equations, under genericity conditions. ca − zc −1 a 1 − za 2 × a −1 − za 1 − z .

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Theta Series for Quantum Loop Algebras and Yangians

Zhang

2024

Commun. Math. Phys.

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Baxter operators and asymptotic representations

Cited by 7 publications

References 41 publications

Elliptic quantum groups and Baxter relations

Elliptic quantum groups and Baxter relations

Length-Two Representations of Quantum Affine Superalgebras and Baxter Operators

Theta Series for Quantum Loop Algebras and Yangians

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