Bosonic realization of toroidal Lie algebras of classical types

Authors:
Naihuan Jing, Kailash C. Misra and Chongbin Xu

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3609-3618

MSC (2000):
Primary 17B60, 17B67, 17B69; Secondary 17A45, 81R10

Published electronically:
June 10, 2009

MathSciNet review:
2529867

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing Feingold and Frenkel's construction, we use Weyl bosonic fields to construct toroidal Lie algebras of types , and of levels and respectively. In particular, our construction also gives new bosonic constructions for orthogonal Lie algebras in the cases of affine Lie algebras.

**[BBS]**Stephen Berman, Yuly Billig, and Jacek Szmigielski,*Vertex operator algebras and the representation theory of toroidal algebras*, Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000) Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 1–26. MR**1919810**, 10.1090/conm/297/05090**[B]**Yuly Billig,*Principal vertex operator representations for toroidal Lie algebras*, J. Math. Phys.**39**(1998), no. 7, 3844–3864. MR**1630546**, 10.1063/1.532472**[FF]**Alex J. Feingold and Igor B. Frenkel,*Classical affine algebras*, Adv. in Math.**56**(1985), no. 2, 117–172. MR**788937**, 10.1016/0001-8708(85)90027-1**[FJW]**Igor B. Frenkel, Naihuan Jing, and Weiqiang Wang,*Vertex representations via finite groups and the McKay correspondence*, Internat. Math. Res. Notices**4**(2000), 195–222. MR**1747618**, 10.1155/S107379280000012X**[G]**Yun Gao,*Fermionic and bosonic representations of the extended affine Lie algebra 𝔤𝔩_{𝔑}(ℂ_{𝕢})*, Canad. Math. Bull.**45**(2002), no. 4, 623–633. Dedicated to Robert V. Moody. MR**1941230**, 10.4153/CMB-2002-057-3**[JMg]**Jiang Cuipo and Meng Daoji,*Vertex representations for the 𝜈+1-toroidal Lie algebra of type 𝐵_{𝑙}*, J. Algebra**246**(2001), no. 2, 564–593. MR**1872115**, 10.1006/jabr.2001.8822**[JM]**N. Jing, K. C. Misra,*Fermionic realization of toroidal Lie algebras of types ABD*, arXiv:0807.3056.**[K]**Victor Kac,*Vertex algebras for beginners*, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. MR**1417941****[L]**Michael Lau,*Bosonic and fermionic representations of Lie algebra central extensions*, Adv. Math.**194**(2005), no. 2, 225–245. MR**2139913**, 10.1016/j.aim.2004.06.005**[MRY]**S. Eswara Rao, R. V. Moody, and T. Yokonuma,*Lie algebras and Weyl groups arising from vertex operator representations*, Nova J. Algebra Geom.**1**(1992), no. 1, 15–57. MR**1163780****[T]**Shaobin Tan,*Vertex operator representations for toroidal Lie algebra of type 𝐵_{𝑙}*, Comm. Algebra**27**(1999), no. 8, 3593–3618. MR**1699582**, 10.1080/00927879908826650

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
17B60,
17B67,
17B69,
17A45,
81R10

Retrieve articles in all journals with MSC (2000): 17B60, 17B67, 17B69, 17A45, 81R10

Additional Information

**Naihuan Jing**

Affiliation:
School of Sciences, South China University of Technology, Guangzhou 510640, People’s Republic of China – and – Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Email:
jing@math.ncsu.edu

**Kailash C. Misra**

Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205

Email:
misra@math.ncsu.edu

**Chongbin Xu**

Affiliation:
School of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, People’s Republic of China

Email:
xuchongbin1977@126.com

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09942-0

Keywords:
Toroidal algebras,
Weyl algebras,
vertex operators,
representations

Received by editor(s):
November 19, 2008

Received by editor(s) in revised form:
February 23, 2009

Published electronically:
June 10, 2009

Additional Notes:
The first author was supported by NSA grant H98230-06-1-0083 and NSFC grant 10728102, and the second author was supported by NSA grant H98230-08-0080.

Communicated by:
Gail R. Letzter

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.