Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras
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Abstract
Abstract 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[ G 0 ] of Vir[ G ] for a direct summand G 0 of G with G = G 0 ⊕ ℤ b , where b ∈ G \ G 0 . This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.
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