-
Previous Article
Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product
- ERA Home
- This Issue
-
Next Article
Tori can't collapse to an interval
Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $
School of Mathematical Sciences, Tongji University, Shanghai 200092, China |
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) $.
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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. 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. |
[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] |
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. |
[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] |
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. |
[18] |
L. Yuan and H. Wu,
Cohomology of the Heisenberg-Virasoro conformal algebra, J. Lie Theory, 26 (2016), 1187-1197.
|
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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. 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. |
[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] |
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. |
[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] |
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. |
[18] |
L. Yuan and H. Wu,
Cohomology of the Heisenberg-Virasoro conformal algebra, J. Lie Theory, 26 (2016), 1187-1197.
|
[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. |
[1] |
Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124 |
[2] |
Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307 |
[3] |
Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086 |
[4] |
Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097 |
[5] |
Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, , () : -. doi: 10.3934/era.2021008 |
[6] |
Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004 |
[7] |
Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006 |
[8] |
P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 |
[9] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[10] |
Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 |
[11] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[12] |
Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 |
[13] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
[14] |
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 |
[15] |
Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029 |
[16] |
Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270 |
[17] |
Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020 doi: 10.3934/jdg.2020033 |
[18] |
Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045 |
[19] |
Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132 |
[20] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
Impact Factor: 0.263
Tools
Article outline
[Back to Top]