On the Jordan kernel of a universal enveloping algebra

Abstract

First it is shown that the Jordan kernel Jk(D(L)) of the division ring of quotients D(L) of the universal enveloping algebra U(L) of a finite-dimensional Lie algebra L is isomorphic to the group ring of the free Abelian group of weights of L in D(L) over a Weyl algebra A,(Z), where Z is a polynomial ring over the center Z(D(L)) of D(L). In particular, it is a maximal order in its division ring of quotients and its center is a unique factorization domain. These two properties are then investigated for the Jordan kernel Jk( U(L)) of U(L). Therefore, the centralizer C,(L”) of the characteristic ideal L” of L in U(L) is studied, since it coincides with Jk(U(L)) in most cases. In particular, we compute its Gelfand-Kirillov dimension and we derive necessary and sufGent conditions in order for CU(Lm) to coincide with Z( U(L”)). Finally, we sharpen the previous results in the case that L is a Frobenius Lie algebra.