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), 36093618
MSC (2000):
Primary 17B60, 17B67, 17B69; Secondary 17A45, 81R10
Published electronically:
June 10, 2009
MathSciNet review:
2529867
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Abstract 
References 
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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, field theory (Charlottesville, VA, 2000)
Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002,
pp. 1–26. MR 1919810
(2003j:17037), http://dx.doi.org/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
(99j:17041), http://dx.doi.org/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
(87d:17018), http://dx.doi.org/10.1016/00018708(85)900271
 [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
(2001c:17042), http://dx.doi.org/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
(2003h:17030), http://dx.doi.org/10.4153/CMB20020573
 [JMg]
Jiang
Cuipo and Meng
Daoji, Vertex representations for the 𝜈+1toroidal Lie
algebra of type 𝐵_{𝑙}, J. Algebra 246
(2001), no. 2, 564–593. MR 1872115
(2003b:17036), http://dx.doi.org/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
(99a:17027)
 [L]
Michael
Lau, Bosonic and fermionic representations of Lie algebra central
extensions, Adv. Math. 194 (2005), no. 2,
225–245. MR 2139913
(2005m:17008), http://dx.doi.org/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
(93h:17040)
 [T]
Shaobin
Tan, Vertex operator representations for toroidal Lie algebra of
type 𝐵_{𝑙}, Comm. Algebra 27 (1999),
no. 8, 3593–3618. MR 1699582
(2001a:17041), http://dx.doi.org/10.1080/00927879908826650
 [BBS]
 S. Berman, Y. Billig, J. Szmigielski, Vertex operator algebras and the representation theory of toroidal algebras. Recent developments in infinitedimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000), 126, Contemp. Math., 297, Amer. Math. Soc., Providence, RI, 2002. MR 1919810 (2003j:17037)
 [B]
 Y. Billig, Principal vertex operator representations for toroidal Lie algebras. J. Math. Phys. 39 (1998), no. 7, 38443864. MR 1630546 (99j:17041)
 [FF]
 A. Feingold, I. B. Frenkel, Classical affine algebras. Adv. Math. 56 (1985), 117172. MR 788937 (87d:17018)
 [FJW]
 I. Frenkel, N. Jing, W. Wang, Vertex representations via finite groups and the McKay correspondence. Int. Math. Res. Notices 4 (2000), 195222. MR 1747618 (2001c:17042)
 [G]
 Y. Gao, Fermionic and bosonic representations of the extended affine Lie algebra . Canad. Math. Bull. 45 (2002), no. 4, 623633. MR 1941230 (2003h:17030)
 [JMg]
 C. Jiang, D. Meng, Vertex representations for the toroidal Lie algebra of type . J. Algebra 246 (2001), no. 2, 564593. MR 1872115 (2003b:17036)
 [JM]
 N. Jing, K. C. Misra, Fermionic realization of toroidal Lie algebras of types ABD, arXiv:0807.3056.
 [K]
 V. G. Kac, Vertex algebras for beginners. Univ. Lecture Ser., 10, Amer. Math. Soc., Providence, RI, 1997. MR 1417941 (99a:17027)
 [L]
 M. Lau, Bosonic and fermionic representations of Lie algebra central extensions. Adv. Math. 194 (2005), no. 2, 225245. MR 2139913 (2005m:17008)
 [MRY]
 R. V. Moody, S. E. Rao, T. Yokonuma, Lie algebras and Weyl groups arising from vertex operator representations. Nova J. Algebra Geom. 1 (1992), no. 1, 1557. MR 1163780 (93h:17040)
 [T]
 S. Tan, Vertex operator representations for toroidal Lie algebra of type . Comm. Algebra 27 (1999), no. 8, 35933618. MR 1699582 (2001a:17041)
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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 276958205
Email:
jing@math.ncsu.edu
Kailash C. Misra
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 276958205
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/S0002993909099420
PII:
S 00029939(09)099420
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 H982300610083 and NSFC grant 10728102, and the second author was supported by NSA grant H98230080080.
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.
