### Electronic Research Archive

2021, Issue 3: 2445-2456. doi: 10.3934/era.2020123
Special Issues

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

• Received: 01 May 2020 Revised: 01 October 2020 Published: 24 November 2020
• 17B10, 17B65, 17B68

• 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 $\mathcal{W}(a,b,r)$.

Citation: Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$[J]. Electronic Research Archive, 2021, 29(3): 2445-2456. doi: 10.3934/era.2020123

### Related Papers:

• 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 $\mathcal{W}(a,b,r)$.

 [1] Cohomology of conformal algebras. Comm. Math. Phys. (1999) 200: 561-598. [2] Poisson vertex algebras in the theory of Hamiltonian equations. Jpn. J. Math. (2009) 4: 141-252. [3] Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B (1984) 241: 333-380. [4] Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. USA (1986) 83: 3068-3071. [5] Conformal modules. Asian J. Math. (1997) 1: 181-193. [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. [7] Structure theory of finite conformal algebras. Selecta Math. (N.S.) (1998) 4: 377-418. [8] Lie conformal algebra cohomology and the variational complex. Comm. Math. Phys. (2009) 292: 667-719. [9] V. Kac, Vertex Algebras for Beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/ulect/010 [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. [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. [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 [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 [14] Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $\mathcal{W}(a, b)$. Comm. Alg. (2018) 46: 5381-5398. [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 [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 [17] $W(a, b)$ Lie conformal algebra and its conformal module of rank one. Alg. Colloq. (2015) 22: 405-412. [18] Cohomology of the Heisenberg-Virasoro conformal algebra. J. Lie Theory (2016) 26: 1187-1197. [19] Structures of $W(2, 2)$ Lie conformal algebra. Open Math. (2016) 14: 629-640.
• © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142