RESEARCH ARTICLE

A class of Lie algebras arising from intersection matrices

  • Li-meng XIA , 1 ,
  • Naihong HU 2
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  • 1. Faculty of Science, Jiangsu University, Zhenjiang 212013, China
  • 2. Department of Mathematics, East China Normal University, Shanghai 200241, China

Received date: 09 May 2013

Accepted date: 19 Jul 2014

Published date: 30 Dec 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix algebras. Moreover, such Lie algebras generated by semi-positive definite matrices can be classified by the modified Dynkin diagrams.

Cite this article

Li-meng XIA , Naihong HU . A class of Lie algebras arising from intersection matrices[J]. Frontiers of Mathematics in China, 2015 , 10(1) : 185 -198 . DOI: 10.1007/s11464-014-0418-y

1
Allison B, Benkart G, Gao Y. Lie Algebras Graded by the Root System BCr, r≥2. Mem Amer Math Soc, No 751. Providence: Amer Math Soc, 2002

2
Benkart G, Zelmanov E. Lie algebras graded by finite root systems and intersection matrix algebras. Invent Math, 1996, 126: 1-45

DOI

3
Berman S. On generators and relations for certain involutory subalgebras of Kac-Moody Lie algebras. Comm Algebra, 1989, 17: 3165-3185

DOI

4
Berman S, Jurisich E, Tan S. Beyond Borcherds Lie algebras and inside. Trans Amer Math Soc, 2001, 353: 1183-1219

DOI

5
Berman S, Moody R V. Lie algebras graded by finite root systems and the intersection matrix algebras of Slowdowy. Invent Math, 1992, 108: 323-347

DOI

6
Bhargava S, Gao Y. Realizations of BCr-graded intersection matrix algebras with grading subalgebras of type Br, r≥3. Pacific J Math, 2013, 263(2): 257-281

DOI

7
Carter R. Lie Algebras of Finite and Affine Type. Cambridge: Cambridge Univ Press, 2005

DOI

8
Eswara Rao S, Moody R V, Yokonuma T. Lie algebras and Weyl groups arising from vertex operator representations. Nova J Algebra and Geometry, 1992, 1: 15-57

9
Gabber O, Kac V G. On defining relations of certain infinite-dimensional Lie algebras. Bull Amer Math Soc (NS), 1981, 5: 185-189

DOI

10
Gao Y. Involutive Lie algebras graded by finite root systems and compact forms of IM algebras. Math Z, 1996, 223: 651-672

DOI

11
Gao Y, Xia L. Finite-dimensional representations for a class of generalized intersection matrix algebras. arXiv: 1404.4310v1

12
Humphreys J E. Introduction to Lie Algebras and Representation Theory. New York: Springer, 1972

DOI

13
Jacobson N. Lie Algebras. New York: Inter Science, 1962

14
Kac V G. Infinite Dimensional Lie Algebras. 3rd ed. Cambridge: Cambridge, Univ Press, 1990

DOI

15
Lusztig G. Introduction to quantum groups. Boston: Birkhäuser, 1993

16
Neher E. Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids. Amer J Math, 1996, 118(2): 439-491

DOI

17
Peng L. Intersection matrix Lie algebras and Ringel-Hall Lie algebras of tilted algebras. In: Happel D, Zhang Y B, eds. Proc 9-th Inter Conf Representation of Algebras, Vol I. Beijing: Beijing Normal Univ Press, 2002, 98-108

18
Peng L, Xu M. Symmetrizable intersection matrices and their root systems. arXiv: 0912.1024

19
Slodowy P. Singularitäaten, Kac-Moody Lie-Algebren, assoziierte Gruppen und Verallgemeinerungen. Habilitationsschrift, Universität Bonn, March 1984

20
Slodowy P. Beyond Kac-Moody algebras and inside. Can Math Soc Conf Proc, 1986, 5: 361-371

21
Xu M, Peng L. Symmetrizable intersection matrix Lie algebras. Algebra Colloq 2011, 18(4): 639-646

DOI

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