On the freeness problem for truncated current algebras

Abstract

Molev (in: Doebner, Scherer, Nattermann (eds) Group 21, physical applications and mathematical aspects of geometry, groups, and algebras, World Scientific, Singapore, vol 1, pp 172–176, 1997) constructed generators of the center of the universal enveloping algebra $$U(\mathfrak{g}_m(n))$$ for the truncated current Lie algebra $$\mathfrak{g}_m(n) = \mathfrak{gl}_n \otimes \mathbb{C}[x]/ (x^m)$$ . Such generators allow to define algebraic varieties associated to the center and the Gelfand–Tsetlin subalgebra. In this paper we prove that the Gelfand–Tsetlin variety is equidimensional of dimension $$mn(n-1)/2$$ if and only if $$n=1,2$$ , implying that $$U(\mathfrak{g}_m(2))$$ is free over the Gelfand–Tsetlin subalgebra.