Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-29T13:57:34.592Z Has data issue: false hasContentIssue false
  • English
  • Français

H-Finite Irreducible Representations of Simple Lie Algebras

Published online by Cambridge University Press:  20 November 2018

F. Lemire  and
M. Pap
F. Lemire
Affiliation:
Université de Montréal, Montréal, Québec
M. Pap
Affiliation:
University of Windsor, Windsor, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 1084 - 1106
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bouwer, I. Z., Standard representations of simple Lie algebras, Can. J. Math. 20 (1968), 344–361.Google Scholar
2. Dixmier, J., Algebres enveloppantes (Gauthier-Villars, Paris, 1974).Google Scholar
3. Gindikin, S. G., Kirillov, A. A. and Fuks, D. B., The works of I. M. GeVfand on functional analysis, algebra and topology, Russian Math. Survey. 29 (1974), 3–61.Google Scholar
4. Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate texts in Mathematics 9 (Springer-Verlag, New York, 1972).Google Scholar
5. Lemire, F. W., Weight spaces and irreducible representations of simple Lie algebras, Proc. Amer. Math. Soc. 22 (1969), 192–197.Google Scholar
6. Lemire, F. W., One-dimensional representation of the cycle subalgebra of a semi-simple Lie algebra, Gan. Math. Bull. 13 (1970), 463–467.Google Scholar
7. Lemire, F. W., A new family of irreducible representations of A n, Can. Math. Bull. 18 (1975), 543–546.Google Scholar