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Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

Published online by Cambridge University Press:  20 November 2018

C. Y. Lee
C. Y. Lee*
Affiliation:
Simon Fraser University, Burnaby, British Columbia
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An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 583 - 592
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Jacobson, N., Lie algebras (Interscience, New Nork, 1962).Google Scholar
2. Harish-Chandra, , Some applications o bra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28–99.Google Scholar
3. Chevalley, , Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.Google Scholar
4. Coxeter, H. S. M., The product of the generators of a finite group generated by reflections, Duke Math. J. 18 (1951), 765–782.Google Scholar
5. Poppov, V. S. and Perelomov, A. M., Casimir operators for semisimple Lie groups, Math. USSR-Izv. 2 (1968), 1313–1335.Google Scholar
6. Humphreys, , Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51–79.Google Scholar
7. Gelfand, I. M., The centre of an infinitesimal group algebra, Mat. Sb. 26 (1950), 103–112.Google Scholar