American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069
[2]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020
[3]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[4]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002
[5]

Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011
[6]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[7]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[8]

Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021046
[9]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
[10]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[11]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021051
[12]

Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080
[13]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
[14]

Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, , () : -. doi: 10.3934/era.2021031
[15]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[16]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[17]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200
[18]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[19]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[20]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (55)
  • HTML views (173)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences