1,671 research outputs found
Finite dimensional representations of W-algebras
W-algebras of finite type are certain finitely generated associative algebras
closely related to the universal enveloping algebras of semisimple Lie
algebras. In this paper we prove a conjecture of Premet that gives an almost
complete classification of finite dimensional irreducible modules for
W-algebras. Also we study a relation between Harish-Chandra bimodules and
bimodules over -algebras.Comment: 19 pages, v2, 21 pages, moderate changes, some mistakes fixed,
Corollary 1.3.3 is added, v3 24 pages three new subsections and several
remarks added v4 proof of Lemma 2.4.1 expanded, Remark 2.3.2 added, v5 29
pages major changes, v6 more changes, v7 fina
Finite W-algebras
A finite W-algebra is an associative algebra constructed from a semisimple
Lie algebra and its nilpotent element. In this survey we review recent
developments in the representation theory of W-algebras. We emphasize various
interactions between W-algebras and universal enveloping algebras.Comment: The text of a sectional talk for ICM 2010. 21 page
1-dimensional representations and parabolic induction for W-algebras
A W-algebra is an associative algebra constructed from a semisimple Lie
algebra and its nilpotent element. This paper concentrates on the study of
1-dimensional representations of these algebras. Under some conditions on a
nilpotent element (satisfied by all rigid elements) we obtain a criterium for a
finite dimensional module to have dimension 1. It is stated in terms of the
Brundan-Goodwin-Kleshchev highest weight theory. This criterium allows to
compute highest weights for certain completely prime primitive ideals in
universal enveloping algebras. We make an explicit computation in a special
case in type . Our second principal result is a version of a parabolic
induction for W-algebras. In this case, the parabolic induction is an exact
functor between the categories of finite dimensional modules for two different
W-algebras. The most important feature of the functor is that it preserves
dimensions. In particular, it preserves one-dimensional representations. A
closely related result was obtained previously by Premet. We also establish
some other properties of the parabolic induction functor.Comment: 31 pages, v2 36 pages, 4 new subsections added, v3 38 pages few gaps
fixed, v4 minor changes, v5 references adde
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