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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dimension formula for graded Lie algebras and its applications
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by Seok-Jin Kang and Myung-Hwan Kim PDF
Trans. Amer. Math. Soc. 351 (1999), 4281-4336 Request permission

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
  • MR Author ID: 307910
  • 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
  • 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.
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4281-4336
  • MSC (1991): Primary 17B01, 17B65, 17B70, 11F22
  • DOI: https://doi.org/10.1090/S0002-9947-99-02239-4
  • MathSciNet review: 1487619