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On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

Published online by Cambridge University Press:  22 January 2016

Hiroshi Kimura
Hiroshi Kimura*
Affiliation:
Tokyo Electrical Engineering College
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Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known;

1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,

2) Determination of all irreducible representations in connection with their highest weights.

3) Weyl’s formula for the character of irreducible representations [9].

4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],

5) The law of the decomposition of the tensor product of two irreducible representations [1].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 211 - 220
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Brauer, R., Sur la multiplication des caractéristiques des groupes continus et semi-simples, C. R. Acad. Sci. Paris, 204 (1937), 1784–1786.Google Scholar
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[4] Hattori, A., On 1-cohomology groups of infinite dimensional representations of semi-simple Lie algebras, J. Math. Soc. Japan, 16 (1964), 226–226.Google Scholar
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[6] Kostant, B., A formula for the multiplicity of a weight, Trans. Amer. Math. Soc, 93 (1959), 53–53.Google Scholar
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[8] Séminaire, “Sophus Lie”, Ecole Normale Supérieure, Paris (1955).Google Scholar
[9] Weyl, H., Über die Darstellungen der halbeinfachen Gruppen durch lineare Transformationen, Math. Z., 24 (1926), 328–328.Google Scholar