The q-characters at roots of unity

We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresponding normalized R-matrices has a pole.

We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters an...

We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.

The q-characters were introduced by Frenkel and mReshetikhin to study finite dimensional representations of the untwisted quantum affine algebra for q generic. The $\epsilon$-characters at roots of unity were constructed by Frenkel and Mukhin to study finite dimensional representations of various specializations of the quantum affine algebra at $q^s=1$. In the finite simply laced case Nakajima defined deformations of q-characters called q,t-characters, for q generic and also ...

In this paper, we will fully describe the representations of the crystallographic rank two affine Hecke algebras using elementary methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.

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 alg...

We classify the irreducible finite-dimensional representations of the twisted quantum affine algebras.

The q-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the q-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the q-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an exa...

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a t...

Here is a list of chapters: 1 Introduction 2 Notation and preliminaries Part I: Finite quantum groups 3 2x2 Matrix quantum groups and the quantum plane 4 Quantum enveloping algebras at a root of unity Part II: q-Oscillators 5 Representations of q-oscillators at a root of unity 6 qr-Oscillator at a root of unity Part III: Infinite quantum groups 7 Quantum affine algebras 8 Quantum affine algebras at a root of unity