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Standard Representations of Simple Lie Algebras

Published online by Cambridge University Press:  20 November 2018

I. Z. Bouwer
I. Z. Bouwer*
Affiliation:
National Research Institute for Mathematical Sciencesy C.S.I.R., Pretoria, South Africa
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Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 344 - 361
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

Based on work done in partial fulfilment of the requirements for the Ph.D. degree at the University of Toronto. The author extends his thanks to the Canadian Mathematical Congress and to the C.S.I.R., Pretoria, for their financial assistance, which enabled him to undertake this research.

References

1. Bargmann, V., Irreducible unitary representations of the Lorentz group, Ann. of Math., 48 1947), 568–640.Google Scholar
2. Chevalley, C., Sur la classification des algèbres de Lie simples et de leurs représentations, Académie des sciences, Séance du 20 novembre 1948, 1136–1138.Google Scholar
3. Harish-Chandra, , On representations of Lie algebras, Ann. of Math., 50, 4 (1949), 900–915.Google Scholar
4. Harish-Chandra, , Some applications of the universal enveloping algebra of a semi-simple Lie algebra, Trans. Amer. Math. Soc, 70 (1951), 28–96.Google Scholar
5. Helgason, S., Differential geometry and symmetric spaces (New York, 1962).Google Scholar
6. Kostant, B., A formula for the multiplicity of a weight, Trans. Amer. Math. Soc, 93 (1959), 53–73.Google Scholar
7. Rickart, C. E., Banach algebras (Princeton, 1960).Google Scholar
8. Rickart, C. E., Séminaire (Sophus Lie“\ Théorie des algèbres de Lie, Topologie des groupes de Lie, 1954/55 Ecole normal supérieure, Paris, 1955).Google Scholar