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doi: 10.3934/era.2020123

Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $

Wenjun Liu , Yukun Xiao and  Xiaoqing Yue ,

School of Mathematical Sciences, Tongji University, Shanghai 200092, China

* Corresponding author: Xiaoqing Yue

Received  May 2020 Revised  October 2020 Published  November 2020

Fund Project: Supported by the NSF grant Nos. 11971350, 11431010 of China and the CSC grant No. 202006260122

We study a family of non-simple Lie conformal algebras $ \mathcal{W}(a,b,r) $ ($ a,b,r\in {\mathbb{C}} $) of rank three with free $ {\mathbb{C}}[{\partial}] $-basis $ \{L, W,Y\} $ and relations $ [L_{\lambda} L] = ({\partial}+2{\lambda})L,\ [L_{\lambda} W] = ({\partial}+ a{\lambda} +b)W,\ [L_{\lambda} Y] = ({\partial}+{\lambda})Y,\ [Y_{\lambda} W] = rW $ and $ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $. In this paper, we investigate the irreducibility of all free nontrivial $ \mathcal{W}(a,b,r) $-modules of rank one over $ {\mathbb{C}}[{\partial}] $ and classify all finite irreducible conformal modules over $ \mathcal{W}(a,b,r) $.

Keywords: Lie conformal algebras, finite conformal modules, irreducible.
Mathematics Subject Classification: 17B10, 17B65, 17B68.
Citation: Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, doi: 10.3934/era.2020123
References:
[1]

B. Bakalov, V. G. Kac and A. A. Voronov, Cohomology of conformal algebras, Comm. Math. Phys., 200 (1999), 561-598.  doi: 10.1007/s002200050541.  Google Scholar

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A. Barakat, A. De Sole and V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Jpn. J. Math., 4 (2009), 141-252.  doi: 10.1007/s11537-009-0932-y.  Google Scholar

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S.-J. Cheng, V. G. Kac and M. Wakimoto, Extensions of conformal modules, in Topological Field Theory, Primitive Forms and Related Topics, Kyoto, (1996), 79–129.  Google Scholar

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A. D'Andrea and V. G. Kac, Structure theory of finite conformal algebras, Selecta Math. (N.S.), 4 (1998), 377-418.  doi: 10.1007/s000290050036.  Google Scholar

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V. G. Kac, The idea of locality, in Physical Application and Mathematical Aspects of Geometry, Groups and Algebras, eds. H.-D. Doebner et al., World Scienctific, Singapore, (1997), 16–32, arXiv: q-alg/9709008v1. Google Scholar

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K. Ling and L. Yuan, Extensions of modules over a class of Lie conformal algebras $\mathcal{W}(b)$, J. Alg. Appl., 18 (2019), 1950164, 13 pp. doi: 10.1142/S0219498819501640.  Google Scholar

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K. Ling and L. Yuan, Extensions of modules over the Heisenberg-Virasoro conformal algebra, Int. J. Math., 28 (2017), 1750036, 13 pp. doi: 10.1142/S0129167X17500367.  Google Scholar

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D. Liu, Y. Hong, H. Zhou and N. Zhang, Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a, b)$, Comm. Alg., 46 (2018), 5381-5398.  doi: 10.1080/00927872.2018.1468903.  Google Scholar

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L. Luo, Y. Hong and Z. Wu, Finite irreducible modules of Lie conformal algebras $\mathcal{W}(a, b)$ and some Schrödinger-Virasoro type Lie conformal algebras, Int. J. Math., 30 (2019), 1950026, 17 pp. doi: 10.1142/S0129167X19500265.  Google Scholar

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H. Wu and L. Yuan, Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra, J. Math. Phys., 58 (2017), 041701, 10 pp. doi: 10.1063/1.4979619.  Google Scholar

[17]

Y. Xu and X. Yue, $W(a, b)$ Lie conformal algebra and its conformal module of rank one, Alg. Colloq., 22 (2015), 405-412.  doi: 10.1142/S1005386715000358.  Google Scholar

[18]

L. Yuan and H. Wu, Cohomology of the Heisenberg-Virasoro conformal algebra, J. Lie Theory, 26 (2016), 1187-1197.   Google Scholar

[19]

L. Yuan and H. Wu, Structures of $W(2, 2)$ Lie conformal algebra, Open Math., 14 (2016), 629-640.  doi: 10.1515/math-2016-0054.  Google Scholar

show all references

References:
[1]

B. Bakalov, V. G. Kac and A. A. Voronov, Cohomology of conformal algebras, Comm. Math. Phys., 200 (1999), 561-598.  doi: 10.1007/s002200050541.  Google Scholar

[2]

A. Barakat, A. De Sole and V. G. Kac, Poisson vertex algebras in the theory of Hamiltonian equations, Jpn. J. Math., 4 (2009), 141-252.  doi: 10.1007/s11537-009-0932-y.  Google Scholar

[3]

A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B, 241 (1984), 333-380.  doi: 10.1016/0550-3213(84)90052-X.  Google Scholar

[4]

R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA, 83 (1986), 3068-3071.  doi: 10.1073/pnas.83.10.3068.  Google Scholar

[5]

S.-J. Cheng and V. G. Kac, Conformal modules, Asian J. Math., 1 (1997), 181-193.  doi: 10.4310/AJM.1997.v1.n1.a6.  Google Scholar

[6]

S.-J. Cheng, V. G. Kac and M. Wakimoto, Extensions of conformal modules, in Topological Field Theory, Primitive Forms and Related Topics, Kyoto, (1996), 79–129.  Google Scholar

[7]

A. D'Andrea and V. G. Kac, Structure theory of finite conformal algebras, Selecta Math. (N.S.), 4 (1998), 377-418.  doi: 10.1007/s000290050036.  Google Scholar

[8]

A. De Sole and V. G. Kac, Lie conformal algebra cohomology and the variational complex, Comm. Math. Phys., 292 (2009), 667-719.  doi: 10.1007/s00220-009-0886-1.  Google Scholar

[9]

V. Kac, Vertex Algebras for Beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/ulect/010.  Google Scholar

[10]

V. G. Kac, The idea of locality, in Physical Application and Mathematical Aspects of Geometry, Groups and Algebras, eds. H.-D. Doebner et al., World Scienctific, Singapore, (1997), 16–32, arXiv: q-alg/9709008v1. Google Scholar

[11]

V. G. Kac, Formal distribution algebras and conformal algebras, in Proc. 12th International Congress Mathematical Physics (ICMP'97)(Brisbane), International Press, Cambridge, (1999), 80–97.  Google Scholar

[12]

K. Ling and L. Yuan, Extensions of modules over a class of Lie conformal algebras $\mathcal{W}(b)$, J. Alg. Appl., 18 (2019), 1950164, 13 pp. doi: 10.1142/S0219498819501640.  Google Scholar

[13]

K. Ling and L. Yuan, Extensions of modules over the Heisenberg-Virasoro conformal algebra, Int. J. Math., 28 (2017), 1750036, 13 pp. doi: 10.1142/S0129167X17500367.  Google Scholar

[14]

D. Liu, Y. Hong, H. Zhou and N. Zhang, Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a, b)$, Comm. Alg., 46 (2018), 5381-5398.  doi: 10.1080/00927872.2018.1468903.  Google Scholar

[15]

L. Luo, Y. Hong and Z. Wu, Finite irreducible modules of Lie conformal algebras $\mathcal{W}(a, b)$ and some Schrödinger-Virasoro type Lie conformal algebras, Int. J. Math., 30 (2019), 1950026, 17 pp. doi: 10.1142/S0129167X19500265.  Google Scholar

[16]

H. Wu and L. Yuan, Classification of finite irreducible conformal modules over some Lie conformal algebras related to the Virasoro conformal algebra, J. Math. Phys., 58 (2017), 041701, 10 pp. doi: 10.1063/1.4979619.  Google Scholar

[17]

Y. Xu and X. Yue, $W(a, b)$ Lie conformal algebra and its conformal module of rank one, Alg. Colloq., 22 (2015), 405-412.  doi: 10.1142/S1005386715000358.  Google Scholar

[18]

L. Yuan and H. Wu, Cohomology of the Heisenberg-Virasoro conformal algebra, J. Lie Theory, 26 (2016), 1187-1197.   Google Scholar

[19]

L. Yuan and H. Wu, Structures of $W(2, 2)$ Lie conformal algebra, Open Math., 14 (2016), 629-640.  doi: 10.1515/math-2016-0054.  Google Scholar

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