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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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  • [1] Richard E. Block, The irreducible representations of the Lie algebra $ {\mathfrak{s}}{\mathfrak{l}}(2)$ and of the Weyl algebra, Adv. in Math. 39 (1981), no. 1, 69-110. MR 605353, https://doi.org/10.1016/0001-8708(81)90058-X
  • [2] Yuly Billig and Kaiming Zhao, Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004), no. 1-2, 23-42. MR 2048305, https://doi.org/10.1016/j.jpaa.2003.12.004
  • [3] N. Chair, V. K. Dobrev, and H. Kanno, $ {\rm SO}(2,{\bf C})$ invariant ring structure of BRST cohomology and singular vectors in $ 2$D gravity with $ c<1$ matter, Phys. Lett. B 283 (1992), no. 3-4, 194-202. MR 1170971, https://doi.org/10.1016/0370-2693(92)90007-Q
  • [4] Hongjia Chen, Xiangqian Guo, and Kaiming Zhao, Tensor product weight modules over the Virasoro algebra, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 829-844. MR 3145133, https://doi.org/10.1112/jlms/jdt046
  • [5] Charles H. Conley and Christiane Martin, A family of irreducible representations of the Witt Lie algebra with infinite-dimensional weight spaces, Compositio Math. 128 (2001), no. 2, 153-175. MR 1850181, https://doi.org/10.1023/A:1017566220585
  • [6] Askar Dzhumadildaev, Virasoro type Lie algebras and deformations, Z. Phys. C 72 (1996), no. 3, 509-517. MR 1415385, https://doi.org/10.1007/s002880050272
  • [7] B. L. Feĭgin and D. B. Fuks, Verma modules over a Virasoro algebra, Funktsional. Anal. i Prilozhen. 17 (1983), no. 3, 91-92. MR 714236
  • [8] Ewa Felińska, Zbigniew Jaskólski, and Michał Kosztołowicz, Whittaker pairs for the Virasoro algebra and the Gaiotto-Bonelli-Maruyoshi-Tanzini states, J. Math. Phys. 53 (2012), no. 3, 033504, 16. MR 2798231, https://doi.org/10.1063/1.3692188
  • [9] Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR 1424041
  • [10] Peter Goddard and David Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Internat. J. Modern Phys. A 1 (1986), no. 2, 303-414. MR 864165, https://doi.org/10.1142/S0217751X86000149
  • [11] Xiangqian Guo, Rencai Lu, and Kaiming Zhao, Fraction representations and highest-weight-like representations of the Virasoro algebra, J. Algebra 387 (2013), 68-86. MR 3056686, https://doi.org/10.1016/j.jalgebra.2013.04.012
  • [12] Kenji Iohara and Yoshiyuki Koga, Representation theory of the Virasoro algebra, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2011. MR 2744610
  • [13] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [14] V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1021978
  • [15] Genqiang Liu, Rencai Lu, and Kaiming Zhao, A class of simple weight Virasoro modules, J. Algebra 424 (2015), 506-521. MR 3293231, https://doi.org/10.1016/j.jalgebra.2014.08.054
  • [16] Rencai Lü, Xiangqian Guo, and Kaiming Zhao, Irreducible modules over the Virasoro algebra, Doc. Math. 16 (2011), 709-721. MR 2861395
  • [17] R. Lü and K. Zhao, A family of simple weight modules over the Virasoro algebra, arXiv:1303.0702.
  • [18] Rencai Lu and Kaiming Zhao, Irreducible Virasoro modules from irreducible Weyl modules, J. Algebra 414 (2014), 271-287. MR 3223399, https://doi.org/10.1016/j.jalgebra.2014.04.029
  • [19] Olivier Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225-234. MR 1144422, https://doi.org/10.1007/BF01231888
  • [20] Volodymyr Mazorchuk and Emilie Wiesner, Simple Virasoro modules induced from codimension one subalgebras of the positive part, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3695-3703. MR 3251711, https://doi.org/10.1090/S0002-9939-2014-12098-3
  • [21] Volodymyr Mazorchuk and Kaiming Zhao, Classification of simple weight Virasoro modules with a finite-dimensional weight space, J. Algebra 307 (2007), no. 1, 209-214. MR 2278050, https://doi.org/10.1016/j.jalgebra.2006.05.007
  • [22] Volodymyr Mazorchuk and Kaiming Zhao, Simple Virasoro modules which are locally finite over a positive part, Selecta Math. (N.S.) 20 (2014), no. 3, 839-854. MR 3217463, https://doi.org/10.1007/s00029-013-0140-8
  • [23] Jonathan Nilsson, Simple $ \mathfrak{sl}_{n+1}$-module structures on $ \mathcal {U}(\mathfrak{h})$, J. Algebra 424 (2015), 294-329. MR 3293222, https://doi.org/10.1016/j.jalgebra.2014.09.036
  • [24] Jonathan Nilsson, $ \mathcal {U}(\mathfrak{h})$-free modules and coherent families, J. Pure Appl. Algebra 220 (2016), no. 4, 1475-1488. MR 3423459, https://doi.org/10.1016/j.jpaa.2015.09.013
  • [25] Matthew Ondrus and Emilie Wiesner, Whittaker modules for the Virasoro algebra, J. Algebra Appl. 8 (2009), no. 3, 363-377. MR 2535995, https://doi.org/10.1142/S0219498809003370
  • [26] Matthew Ondrus and Emilie Wiesner, Whittaker categories for the Virasoro algebra, Comm. Algebra 41 (2013), no. 10, 3910-3930. MR 3169498, https://doi.org/10.1080/00927872.2012.693557
  • [27] Haijun Tan and Kaiming Zhao, Irreducible Virasoro modules from tensor products (II), J. Algebra 394 (2013), 357-373. MR 3092725, https://doi.org/10.1016/j.jalgebra.2013.07.023
  • [28] Haijun Tan and Kaiming Zhao, $ \mathcal {W}_n^+$- and $ \mathcal {W}_n$-module structures on $ U(\mathfrak{h}_n)$, J. Algebra 424 (2015), 357-375. MR 3293224, https://doi.org/10.1016/j.jalgebra.2014.09.031
  • [29] Shintarou Yanagida, Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial, J. Algebra 333 (2011), 273-294. MR 2785949, https://doi.org/10.1016/j.jalgebra.2011.02.039
  • [30] Hechun Zhang, A class of representations over the Virasoro algebra, J. Algebra 190 (1997), no. 1, 1-10. MR 1442144, https://doi.org/10.1006/jabr.1996.6565

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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

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