SOME NORMS ON UNIVERSAL ENVELOPING-ALGEBRAS

The universal enveloping algebra, Li(g), of a Lie algebra B supports s ome norms and seminorms that have arisen naturally in the context of h eat kernel analysis on Lie groups. These norms and seminorms are inves tigated here from an algebraic viewpoint. It is shown that the norms c orresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism U(g(1) + g(2)) congruent to U(g(1)) x U(g(2)) The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for s1(2, C). It is also sh own that the algebraic dual space U' is spanned by its finite rank ele ments if and only if fl is nilpotent.