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Finite Dimensional Representations of Quantum Affine Algebras  [PDF]
Michael Kleber
Mathematics , 1998,
Abstract: We investigate the characters of some finite-dimensional representations of the quantum affine algebras $U_q(\hat{g})$ using the action of the copy of $U_q(g)$ embedded in it. First, we present an efficient algorithm for computing the Kirillov-Reshetikhin conjectured formula for these characters when $g$ is simply-laced. This replaces the original formulation, in terms of "rigged configurations", with one based on polygonal paths in the Weyl chamber. It also gives a new algorithm for decomposing a tensor product of any number of representations of $sl(n)$ corresponding to rectangular Young diagrams, in a way symmetric in all the factors. This section is an expanded version of q-alg/9611032 . Second, we study a generalization of certain remarkable quadratic relations that hold among characters of $sl(n)$ (the "discrete Hirota relations") whose solutions seem to be characters of quantum affine algebras. We use show that these relations have a unique solution over characters of $U_q(g)$.
On finite-dimensional representations of two-parameter quantum affine algebras  [PDF]
Naihuan Jing,Honglian Zhang
Mathematics , 2012, DOI: 10.1142/S0219498816500547
Abstract: A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.
Spectral Characters of Finite-Dimensional Representations of Affine Algebras  [PDF]
Vyjayanthi Chari,Adriano Moura
Mathematics , 2003,
Abstract: We introduce the notion of a spectral character for finite-dimensional representations of affine algebras. These can be viewed as a suitable q=1 limit of the elliptic characters defined by Etingof and Moura for quantum affine algebras. We show that these characters determine blocks of the category of finite-dimensional modules for affine algebras. To do this we use the Weyl modules defined by Chari and Pressley and some indecomposable reducible quotient of the Weyl modules.
Finite Dimensional Representations of Quantum Affine Algebras  [PDF]
Gustav W. Delius,Yao-Zhong Zhang
Physics , 1994, DOI: 10.1088/0305-4470/28/7/014
Abstract: We give a general construction for finite dimensional representations of $U_q(\hat{\G})$ where $\hat{\G}$ is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. At $q=1$ this is trivial because $U(\hat{\G})=U({\G})\otimes \C(x,x^{-1})$ with $\G$ a finite dimensional Lie algebra. But this fact no longer holds after quantum deformation. In most cases it is necessary to take the direct sum of several irreducible $U_q({\G})$-modules to form an irreducible $U_q(\hat{\G})$-module which becomes reducible at $q = 1$. We illustrate our technique by working out explicit examples for $\hat{\G}=\hat{C}_2$ and $\hat{\G}=\hat{G}_2$. These finite dimensional modules determine the multiplet structure of solitons in affine Toda theory.
A Family of Finite-dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank  [PDF]
Yuchen Fu,Seth Shelley-Abrahamson
Mathematics , 2015,
Abstract: We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of rank n using quantum groups. Our construction is motivated by the rational GDAHA representations given by Silvia Montarani.
Quiver varieties and finite dimensional representations of quantum affine algebras  [PDF]
Hiraku Nakajima
Mathematics , 1999,
Abstract: We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
t-analogue of the q-characters of finite dimensional representations of quantum affine algebras  [PDF]
Hiraku Nakajima
Mathematics , 2000,
Abstract: Frenkel-Reshetikhin introduced $q$-characters of finite dimensional representations of quantum affine algebras. We give a combinatorial algorithm to compute them for all simple modules. Our tool is $t$-analogue of the $q$-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition.
Elliptic Central Characters and Blocks of Finite Dimensional Representations of Quantum Affine Algebras  [PDF]
Pavel I. Etingof,Adriano A. Moura
Mathematics , 2002,
Abstract: The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a direct sum of indecomposable subcategorie (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of $U_q(\hat{{\mathfrak g}})$. Then we show that the block decomposition of this category is paramentrized by these elliptic central characters.
Finite dimensional representations of quantum affine algebras at roots of unity  [PDF]
Jonathan Beck,Victor G. Kac
Physics , 1994,
Abstract: We describe explicitly the canonical map $\chi:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional irreducible representations of an algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $\chi$ of Spec $\ue(\a{g})$ is the subgroup of principal adeles.
Quantum affine algebras and affine Hecke algebras  [PDF]
Vyjayanthi Chari,Andrew Pressley
Mathematics , 1995,
Abstract: We describe a connection between finite--dimensional representations of quantum affine algebras and affine Hecke algebras.
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