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A class of simple non-weight modules over the virasoro algebra

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  07 August 2020

Haibo Chen  and
JianZhi Han
Haibo Chen
Affiliation:
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai201209, China (rebel1025@126.com)
JianZhi Han
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai200092, China (jzhan@tongji.edu.cn)
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Abstract

The Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis {Lm, C| m ∈ ℤ} and relations [Lm, Ln] = (n − m)Lm+n + δm+n,0((m3 − m)/12)C, [Lm, C] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by Li for i ≥ −1. For any triple (μ, λ, α) of complex numbers with μ ≠ 0, λ ≠ 0 and any non-trivial $\mathfrak a$-module V satisfying the condition: for any v ∈ V there exists a non-negative integer m such that Liv = 0 for all i ≥ m, non-weight $\mathcal {L}$-modules on the linear tensor product of V and ℂ[∂], denoted by $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))\ (\Omega (\lambda ,\alpha )=\mathbb {C}[\partial ]$ as vector spaces), are constructed in this paper. We prove that $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if μ ≠ 1, λ ≠ 0, α ≠ 0. We also give necessary and sufficient conditions for two such simple $\mathcal {L}$-modules being isomorphic. Finally, these simple $\mathcal {L}$-modules $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are proved to be new for V not being the highest weight $\mathfrak a$-module whose highest weight is non-zero.

Keywords

Virasoro algebranon-weight modulesimple module

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B65: Infinite-dimensional Lie (super)algebras 17B68: Virasoro and related algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 956 - 970
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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