Quantum Loop Algebras and ℓ-Root Operators

Young, Charles A. S.

doi:10.1007/s00031-015-9339-4

Cited by 6 publications

(1 citation statement)

References 34 publications

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“…We merely wish to remark that the proofs of this lemma and Theorem 4.1 can be easily modified to work for Yangians and quantum loop algebras. Our proof uses techniques which are familiar in the theory of quantum loop algebras, see e.g., [51]. 4.3.…”

Section: Classification Of Irreducibles I: Necessary Conditionmentioning

confidence: 99%

Elliptic quantum groups and their finite-dimensional representations

Gautam¹,

Toledano-Laredo²

2017

Preprint

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Let g be a complex semisimple Lie algebra, τ a point in the upper half-plane, and ∈ C a deformation parameter such that the image of in the elliptic curve C/(Z + τ Z) is of infinite order. In this paper, we give an intrinsic definition of the category of finite-dimensional representations of the elliptic quantum group E ,τ (g) associated to g. The definition is given in terms of Drinfeld half-currents, and extends that given by Enriquez-Felder for g = sl 2 [14]. When g = sln, it reproduces Felder's RLL definition [24] via the Gauß decomposition obtained in [14] for n = 2 and [31] for n ≥ 3. We classify the irreducible representations of E ,τ (g) in terms of elliptic Drinfeld polynomials, in close analogy to the case of the Yangian Y (g) and quantum loop algebra Uq(Lg) of g. A crucial ingredient in the classification is a functor Θ from finitedimensional representations of Uq(Lg) to those of E ,τ (g), which is an elliptic analogue of the monodromy functor constructed in [33], and circumvents the fact that E ,τ (g) does not admit Verma modules. Our classification is new even for g = sl 2 . It holds more generally when g is a symmetrisable Kac-Moody algebra, provided finite-dimensionality is replaced by an integrability and category O conditions.

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Section: Classification Of Irreducibles I: Necessary Conditionmentioning

confidence: 99%

Elliptic quantum groups and their finite-dimensional representations

Gautam¹,

Toledano-Laredo²

2017

Preprint

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Finite Type Modules and Bethe Ansatz Equations

Feigin¹,

Jimbo²,

Miwa³

et al. 2017

Ann. Henri Poincaré

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Abstract. We introduce and study a category O f in b of modules of the Borel subalgebra U q b of a quantum affine algebra U q g, where the commutative algebra of Drinfeld generators h i,r , corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional U q g modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in O f in b . Among them we find the Baxter Q i operators and T i operators satisfying relations of the form T i Q i = j Q j + k Q k . We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the Q i operators acting in an arbitrary finite-dimensional representation of U q g.

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Poles of finite-dimensional representations of Yangians

Gautam

Wendlandt

2022

Sel. Math. New Ser.

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Quantum Loop Algebras and ℓ-Root Operators

Cited by 6 publications

References 34 publications

Elliptic quantum groups and their finite-dimensional representations

Elliptic quantum groups and their finite-dimensional representations

Finite Type Modules and Bethe Ansatz Equations

Poles of finite-dimensional representations of Yangians

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