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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dimension formula for graded Lie algebras
and its applications


Authors: Seok-Jin Kang and Myung-Hwan Kim
Journal: Trans. Amer. Math. Soc. 351 (1999), 4281-4336
MSC (1991): Primary 17B01, 17B65, 17B70, 11F22
Published electronically: June 29, 1999
MathSciNet review: 1487619
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the structure of infinite dimensional Lie algebras $L=\bigoplus _{\alpha \in \Gamma } L_{\alpha }$ graded by a countable abelian semigroup $\Gamma $ satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the $\Gamma $-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces $L_{\alpha }$ $(\alpha \in \Gamma )$. Our dimension formula enables us to study the structure of the $\Gamma $-graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.


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Additional Information

Seok-Jin Kang
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Email: sjkang@math.snu.ac.kr

Myung-Hwan Kim
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
Email: mhkim@math.snu.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02239-4
PII: S 0002-9947(99)02239-4
Received by editor(s): May 23, 1997
Published electronically: June 29, 1999
Additional Notes: This research was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1996.
Article copyright: © Copyright 1999 American Mathematical Society