A class of simple non-weight modules over the virasoro algebra. Issue 4 (7th November 2020)

• Record Type: Journal Article
• Authors: Chen, Haibo; Han, JianZhi
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 63:Issue 4(2020)
• Page Start: 956
• Page End: 970
• Publication Date: 2020-11-07
• Subjects: Virasoro algebra, -- non-weight module, -- simple module; 17B10, -- 17B65, -- 17B68; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091520000279
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: The Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis { L m, C | m ∈ ℤ} and relations [ L m, L n ] = ( n − m ) L m + n + δ m + n, 0 (( m 3 − m )/12) C, [ L m, C ] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by L i 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 L i v = 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.