Category O for Quantum Groups
نویسنده
چکیده
In this paper we study the BGG-categories Oq associated to quantum groups. We prove that many properties of the ordinary BGG-category O for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for O and for finite dimensional Uq-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in Oq. As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category O.
منابع مشابه
ar X iv : q - a lg / 9 50 30 22 v 1 3 1 M ar 1 99 5 Representations of Quantum Affine Algebras
Introduction Let G be a finite dimensional simple Lie algebra and G the corresponding affine Kač-Moody algebra. The notion of the fusion in the category O of representations of affine Kač-Moody algebras G was introduced ten years ago by physicists in the framework of Conformal Field Theory. This notion was developed in a number of mathematical papers (see, for example, [TUY]), where the notion ...
متن کاملAffinization of Category O for Quantum Groups
Let g be a simple Lie algebra. We consider the category Ô of those modules over the affine quantum group Uq(ĝ) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category Ô. In particular, we develop the theory of q-characters a...
متن کاملBraided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras
Braided Lie algebras and bicovariant differential calculi over co-quasitriangular Hopf algebras. Abstract We show that if g Γ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a co-quasitriangular Hopf algebra (A, r), then a certain extension of it is a braided Lie algebra in the category of A-comodules. This...
متن کاملCrossed Modules and Quantum Groups in Braided Categories
Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category DY (C)AA of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group (A,A,R) the correspo...
متن کاملSubalgebras of Quasi-hereditary Algebras Arising from Algebraic and Quantum Groups
Quasi-hereditary algebras arise in several natural contexts. For example, their module categories often appear in connection with the representation theory of algebraic groups over elds of positive characteristic or of quantum groups at a root of unity. This point of view has been exploited by various authors, e. g., [CPS3{8], [G2], etc. Classical Schur algebras attached to the general linear g...
متن کامل