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On basic Cycles of An, Bn, Cn and Dn

Published online by Cambridge University Press:  20 November 2018

D. J. Britten  and
F. W. Lemire
D. J. Britten
Affiliation:
University of Windsor, Windsor, Ontario
F. W. Lemire
Affiliation:
University of Windsor, Windsor, Ontario
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In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.

First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 1 , 01 February 1985 , pp. 122 - 140
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Britten, D.J. and Lemire, F. W., Irreducible representations of An with a 1-dimensional weight space, Trans. Amer. Math. Soc. 273 (1982), 509–540.Google Scholar
2. Dixmier, J., Sur les homomorphismes d'Harish-Chandra, Inventiones Math. 17 (1972), 167–176.Google Scholar
3. Humphreys, J., Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9 (Springer-Verlag, New York, 1972).CrossRefGoogle Scholar
4. van den Hombergh, A., Sur des suites de racines dont la sommes des termes est nulle, Bull. Soc. Math. France 102 (1974), 353–364.Google Scholar