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 m(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 U(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 sufficient conditions in order for C U(L ∞) to coincide with Z(U(L ∞)) . Finally, we sharpen the previous results in the case that L is a Frobenius Lie algebra.