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 = {α 1, ..., α n } a base of Φ, and Φ+ = {β 1, ..., β m } the positive roots corresponding to Δ.
Britten, D. J.; Lemire, F. W. On basic Cycles of An , Bn , Cn and Dn. Canadian journal of mathematics, Tome 37 (1985) no. 1, pp. 122-140. doi: 10.4153/CJM-1985-010-8
@article{10_4153_CJM_1985_010_8,
author = {Britten, D. J. and Lemire, F. W.},
title = {On basic {Cycles} of {An} , {Bn} , {Cn} and {Dn}},
journal = {Canadian journal of mathematics},
pages = {122--140},
year = {1985},
volume = {37},
number = {1},
doi = {10.4153/CJM-1985-010-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1985-010-8/}
}
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AU - Britten, D. J.
AU - Lemire, F. W.
TI - On basic Cycles of An , Bn , Cn and Dn
JO - Canadian journal of mathematics
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EP - 140
VL - 37
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