Classification of irreducible weight modules

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

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Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091510001604 ◽

2012 ◽

Vol 55 (3) ◽

pp. 697-709 ◽

Cited By ~ 4

Author(s):

Xiangqian Guo ◽

Rencai Lu ◽

Kaiming Zhao

Keyword(s):

Direct Summand ◽

Virasoro Algebra ◽

Additive Subgroup ◽

One Dimensional ◽

Finite Dimensional ◽

Irreducible Modules ◽

Intermediate Series ◽

Complex Number Field ◽

Weight Modules

AbstractLet 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.

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Supports of weight modules over Witt algebras

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210509000912 ◽

2011 ◽

Vol 141 (1) ◽

pp. 155-170 ◽

Cited By ~ 19

Author(s):

Volodymyr Mazorchuk ◽

Kaiming Zhao

Keyword(s):

Lie Algebra ◽

Weight Module ◽

Root Space ◽

Abelian Subalgebra ◽

Space Decomposition ◽

Finite Dimensional ◽

Graded Lie Algebra ◽

Alpha Beta ◽

Weight Modules

As the first step towards a classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe the supports of such modules. We also obtain some descriptions of the support of an arbitrary simple weight module over a ℤn-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\smash{\bigoplus_{\alpha\in\mathbb{Z}^n}\mathfrak{g}_\alpha}$ with respect to the abelian subalgebra $\mathfrak{g}_0$, with the property $\smash{[\mathfrak{g}_\alpha,\mathfrak{g}_\beta] = \mathfrak{g}_{\alpha+\beta}}$ for all α, β ∈ ℤn, α ≠ β (this class contains the algebra Wn).

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