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Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  12 April 2012

Xiangqian Guo ,
Rencai Lu  and
Kaiming Zhao
Xiangqian Guo
Affiliation:
Department of Mathematics, Zhengzhou university, Zhengzhou 450001, Henan, People's Republic of China (guoxq@amss.ac.cn)
Rencai Lu
Affiliation:
Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu, People's Republic of China (rencail@amss.ac.cn)
Kaiming Zhao
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada (kzhao@wlu.ca) and College of Mathematics and Information Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei 050016, People's Republic of China
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Abstract

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Let G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

Keywords

generalized Virasoro algebraweight moduleHarish-Chandra module

MSC classification

Secondary: 17B10: Representations, algebraic theory (weights) 17B20: Simple, semisimple, reductive (super)algebras 17B65: Infinite-dimensional Lie (super)algebras 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B68: Virasoro and related algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 697 - 709
Copyright
Copyright © Edinburgh Mathematical Society 2012

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