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Generalized polynomial modules over the Virasoro algebra


Authors: Genqiang Liu and Yueqiang Zhao
Journal: Proc. Amer. Math. Soc. 144 (2016), 5103-5112
MSC (2010): Primary 17B10, 17B65, 17B66, 17B68
DOI: https://doi.org/10.1090/proc/13171
Published electronically: June 10, 2016
MathSciNet review: 3556256
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Abstract: Let $ \mathcal {B}_r$ be the $ (r+1)$-dimensional quotient Lie algebra of the positive part of the Virasoro algebra $ \mathcal {V}$. Irreducible $ \mathcal {B}_r$-modules were used to construct irreducible Whittaker modules in a work of Mazorchuk and Zhao (2014) and irreducible weight modules with infinite dimensional weight spaces over $ \mathcal {V}$ in a work of Liu, Lu and Zhao (2015). In the present paper, we construct non-weight Virasoro modules $ F(M, \Omega (\lambda ,\beta ))$ from irreducible $ \mathcal {B}_r$-modules $ M$ and $ (\mathcal {A},\mathcal {V})$-modules $ \Omega (\lambda ,\beta )$. We give necessary and sufficient conditions for the Virasoro module $ F(M, \Omega (\lambda ,\beta ))$ to be irreducible. Using the weighting functor introduced by J. Nilsson, we also determine necessary and sufficient conditions for two $ F(M, \Omega (\lambda ,\beta ))$ to be isomorphic.


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Additional Information

Genqiang Liu
Affiliation: School of Mathematics and Statistics, Henan University, Kaifeng 475004, People’s Republic of China
Email: liugenqiang@amss.ac.cn

Yueqiang Zhao
Affiliation: School of Mathematics and Statistics, Henan University, Kaifeng 475004, People’s Republic of China
Email: yueqiangzhao@163.com

DOI: https://doi.org/10.1090/proc/13171
Keywords: Virasoro algebra, non-weight modules, irreducible modules, weighting functor
Received by editor(s): December 16, 2015
Received by editor(s) in revised form: February 16, 2016
Published electronically: June 10, 2016
Additional Notes: The first author was supported in part by the NSF of China (Grant 11301143) and grants at Henan University (2012YBZR031, yqpy20140044).
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2016 American Mathematical Society