A class of simple non-weight modules over the virasoro algebra
A class of simple non-weight modules over the virasoro algebra
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DOI：
10.1017/s0013091520000279
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发表时间：
2020-08
影响因子：
0.7
通讯作者：
Haibo Chen;Jianzhi Han
中科院分区：
文献类型：
--
作者：
Haibo Chen;Jianzhi Han
Virasoro代数$\mathcal {L}$是一个以基{Lm，C| m ∈ N}和关系[Lm，Ln] =（n-m）Lm+n + δm+n，0（（m3-m）/12）C，对于m，n ∈ N，[Lm，C] = 0.设$\mathfrak a$是Li张成的$\mathcal {L}$的子代数，其中i ≥ −1。对于任何三元组（μ，λ，α）的复数与μ 0，λ 0和任何非平凡$\mathfrak $-模V满足条件：对任意v ∈ V，存在一个非负整数m使得Li v = 0，对所有i ≥ m，V和V的线性张量积上的非权$\mathcal {L}$-模，记为$\mathcal {M}（V，\mu，\Omega（\lambda，\alpha））\（\Omega（\lambda，\alpha）=\mathbb {C}[\partial ]$作为向量空间）。我们证明了$\mathcal {M}（V，\mu，\Omega（\lambda，\alpha））$是单的当且仅当μ <$1，λ <$0，α <$0。我们还给出了两个这样的简单$\mathcal {L}$-模同构的充分必要条件。最后，证明了这些简单$\mathcal {L}$-模$\mathcal {M}（V，\mu，\Omega（\lambda，\alpha））$是新的，因为V不是最高权$\mathfrak一个最高权非零的$-模。
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 Li 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.