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
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.
|
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
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