On the algebra of representative functions of a Lie algebra

Abstract

1. Introduction. This paper deals with the structure of the algebra of representative functions for Lie algebras. We shall be concerned mostly with proving anialogs of certain of the results which are known from [5] for the case of Lie groups. Our results may also be viewed as a natural extension of [2] and [3]. More specifially, in ??3 and 4, we treat a special class of subalgebras of R(L), the algebra of representative functions on the universal enveloping algebra of a Lie algebra L, called normal basic subalgebras, which are fundamental for the representation theory. For example, we prove that the semisimple part of such a subalgebra is always the same, and prove a conjugacy theorem for these subalgebras. In ?5, we turn our attention to the group of the proper automorphisms of R(L). We analyze its intrinsic structure, and also obtain this group as the inverse limit of its restriction images on the finite-dimensional stable subspaces of R(L). These considerations lead naturally to the discussion of a more general inverse limit system, namely of irreducible algebraic linear groups and rational group epimorphisms, which we undertake in ?6. In conclusion, the auther wishes to thank Professor Hochschild for his invaluable assistance in the writing of this thesis.