Dimension formula for graded Lie algebras and its applications

Kang, Seok-Jin; Kim, Myung-Hwan

doi:10.1090/S0002-9947-99-02239-4

Dimension formula for graded Lie algebras and its applications
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by Seok-Jin Kang and Myung-Hwan Kim

Trans. Amer. Math. Soc. 351 (1999), 4281-4336

DOI: https://doi.org/10.1090/S0002-9947-99-02239-4

Published electronically: June 29, 1999

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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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