Computing invariants and semi-invariants by means of Frobenius Lie algebras

Abstract

Let U(g) be the enveloping algebra of a finite-dimensional Lie algebra g over a field k of characteristic zero, Z(U(g)) its center and Sz(U(g)) its semi-center. A sufficient condition is given in order for Sz(U(g)) to be a polynomial algebra over k. Surprisingly, this condition holds for many Lie algebras, especially among those for which the radical is nilpotent, in which case Sz(U(g)) = Z(U(g)). In particular, it allows the explicit description of Z(U(g)) for more than half of all complex, indecomposable nilpotent Lie algebras of dimension at most 7. (C) 2008 Elsevier Inc. All rights reserved.