On certain commutative subalgebras of a universal enveloping algebra

Abstract

The question is posed of lifting, to the universal enveloping algebra of a semisimple Lie algebra, the commutative subalgebras constructed by A. S. Mishchenko and A. T. Fomenko in the corresponding Poisson-Lie-Berezin algebra by the transla- tion-of-invariants method. It is shown that such a lifting is in any case possible up to degree 2. A characterization is given of the commutative subspaces of elements of degree 2 so obtained in the universal enveloping algebra, as well as a description of the subspaces obtained from them by a limit passage. In the case of the algebra of all matrices with zero trace, a connection is exhibited between one of these limit subspaces and the construction of a Geťfand-Tsetlin basis. © 1991 IOP Publishing Ltd.