1,998 research outputs found
Extended T-systems
We use the theory of q-characters to establish a number of short exact
sequences in the category of finite-dimensional representations of the quantum
affine groups of types A and B. That allows us to introduce a set of 3-term
recurrence relations which contains the celebrated T-system as a special case.Comment: 36 pages, latex; v2: version to appear in Selecta Mathematic
On multigraded generalizations of Kirillov-Reshetikhin modules
We study the category of Z^l-graded modules with finite-dimensional graded
pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre
subcategories with finitely many isomorphism classes of simple objects. We
construct projective resolutions for the simple modules in these categories and
compute the Ext groups between simple modules. We show that the projective
covers of the simple modules in these Serre subcategories can be regarded as
multigraded generalizations of Kirillov-Reshetikhin modules and give a
recursive formula for computing their graded characters
On minimal affinizations of representations of quantum groups
In this paper we study minimal affinizations of representations of quantum
groups (generalizations of Kirillov-Reshetikhin modules of quantum affine
algebras introduced by Chari). We prove that all minimal affinizations in types
A, B, G are special in the sense of monomials. Although this property is not
satisfied in general, we also prove an analog property for a large class of
minimal affinization in types C, D, F. As an application, the Frenkel-Mukhin
algorithm works for these modules. For minimal affinizations of type A, B we
prove the thin property (the l-weight spaces are of dimension 1) and a
conjecture of Nakai-Nakanishi (already known for type A). The proof of the
special property is extended uniformly for more general quantum affinizations
of quantum Kac-Moody algebras.Comment: 38 pages; references and additional results added. Accepted for
publication in Communications in Mathematical Physic
Langlands duality for finite-dimensional representations of quantum affine algebras
We describe a correspondence (or duality) between the q-characters of
finite-dimensional representations of a quantum affine algebra and its
Langlands dual in the spirit of q-alg/9708006 and 0809.4453. We prove this
duality for the Kirillov-Reshetikhin modules and their irreducible tensor
products. In the course of the proof we introduce and construct "interpolating
(q,t)-characters" depending on two parameters which interpolate between the
q-characters of a quantum affine algebra and its Langlands dual.Comment: 40 pages; several results and comments added. Accepted for
publication in Letters in Mathematical Physic
Representations of Double Affine Lie algebras
We study representations of the double affine Lie algebra associated to a
simple Lie algebra. We construct a family of indecomposable integrable
representations and identify their irreducible quotients. We also give a
condition for the indecomposable modules to be irreducible, this is analogous
to a result in the representation theory of quantum affine algebras. Finally,
in the last section of the paper, we show, by using the notion of fusion
product, that our modules are generically reducible
Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras
Let Uq(ghat) be the quantum affine algebra associated to a simply-laced
simple Lie algebra g. We examine the relationship between Dorey's rule, which
is a geometrical statement about Coxeter orbits of g-weights, and the structure
of q-characters of fundamental representations V_{i,a} of Uq(ghat). In
particular, we prove, without recourse to the ADE classification, that the rule
provides a necessary and sufficient condition for the monomial 1 to appear in
the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.Comment: 30 pages, latex; v2, to appear in Communications in Mathematical
Physic
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