## Bosonic realization of toroidal Lie algebras of classical types

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- by Naihuan Jing, Kailash C. Misra and Chongbin Xu
- Proc. Amer. Math. Soc.
**137**(2009), 3609-3618 - DOI: https://doi.org/10.1090/S0002-9939-09-09942-0
- Published electronically: June 10, 2009
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## Abstract:

Generalizing Feingold and Frenkel’s construction, we use Weyl bosonic fields to construct toroidal Lie algebras of types $A_n, B_n$, $C_n$ and $D_n$ of levels $-1, -2, -1/2$ and $-2$ respectively. In particular, our construction also gives new bosonic constructions for orthogonal Lie algebras in the cases of affine Lie algebras.## References

- 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**, DOI 10.1090/conm/297/05090 - Yuly Billig,
*Principal vertex operator representations for toroidal Lie algebras*, J. Math. Phys.**39**(1998), no. 7, 3844–3864. MR**1630546**, DOI 10.1063/1.532472 - Alex J. Feingold and Igor B. Frenkel,
*Classical affine algebras*, Adv. in Math.**56**(1985), no. 2, 117–172. MR**788937**, DOI 10.1016/0001-8708(85)90027-1 - 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**, DOI 10.1155/S107379280000012X - Yun Gao,
*Fermionic and bosonic representations of the extended affine Lie algebra $\mathfrak {gl}_N({\Bbb C}_q)$*, Canad. Math. Bull.**45**(2002), no. 4, 623–633. Dedicated to Robert V. Moody. MR**1941230**, DOI 10.4153/CMB-2002-057-3 - Jiang Cuipo and Meng Daoji,
*Vertex representations for the $\nu +1$-toroidal Lie algebra of type $B_l$*, J. Algebra**246**(2001), no. 2, 564–593. MR**1872115**, DOI 10.1006/jabr.2001.8822 - N. Jing, K. C. Misra,
*Fermionic realization of toroidal Lie algebras of types ABD*, arXiv:0807.3056. - Victor Kac,
*Vertex algebras for beginners*, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. MR**1417941**, DOI 10.1090/ulect/010 - Michael Lau,
*Bosonic and fermionic representations of Lie algebra central extensions*, Adv. Math.**194**(2005), no. 2, 225–245. MR**2139913**, DOI 10.1016/j.aim.2004.06.005 - 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** - Shaobin Tan,
*Vertex operator representations for toroidal Lie algebra of type $B_l$*, Comm. Algebra**27**(1999), no. 8, 3593–3618. MR**1699582**, DOI 10.1080/00927879908826650

## Bibliographic 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
- MR Author ID: 232836
- Email: jing@math.ncsu.edu
**Kailash C. Misra**- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
- MR Author ID: 203398
- 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
- 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
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 3609-3618 - MSC (2000): Primary 17B60, 17B67, 17B69; Secondary 17A45, 81R10
- DOI: https://doi.org/10.1090/S0002-9939-09-09942-0
- MathSciNet review: 2529867