Note on faithful representations and a local property of Lie groups

Abstract

Let G be any analytic group, let T be a maximal toroid of the radical of G, and let S be a maximal sernisimnple analytic subgroup of G. If L = L(G) is the Lie algebra of G, rad[L, L] is the radical of [L, L], and Z(L) is the center of L, we show that G has a faithful representation if and only if (i) rad[L, L] n Z(L) n L(T) = (0), and (ii) S has a faithful representation. A theorem of M. Moskowitz [4, Thm. 2], shows that if L is a finite-dimensional (real) Lie algebra, then all analytic groups with Lie algebra L have faithful representations if and only if (i) rad[L, L] n Z(L) = (0), and (ii) for some maximal semisimple subalgebra S of L, the simply connected analytic group with Lie algebra S has a faithful representation. So it would be of interest to find a similar criterion for a single analytic group G to have a faithful representation. Such a criterion is given in Theorem 2 below. As a consequence, we obtain Moskowitz' Theorem in Corollary 3. So our criterion in the solvable case says that G has a faithful representation if and only if [L, L] n Z(L) n L(T) = (0) for some maximal toroid T of G where L = L(G); whereas the well-known criterion in the solvable case is that G has a faithful representation if and only if [G, G] is closed in G and simply connected [2, p. 220]. For the case of semisimple analytic groups, we refer the reader to [2, pp. 199-201]. Our proof uses the notion of nuclei of analytic groups introduced by Hochschild and Mostow. A nucleus K of an analytic group G is a closed normal simply connected solvable analytic subgroup of G such that G/K is reductive. An analytic group G has a faithful representation if and only if G has a nucleus; if G has a nucleus K, then G = K P (semi-direct) for every maximal reductive analytic subgroup P of G [3, Section 2]. Recall that an analytic group is reductive if it has a faithful representation and all its representations are semisimple. If G is an analytic group, L:(G) is its Lie algebra, rad G is its radical, and [G, G] is its commutator (derived) subgroup. Similarly, if L is a Lie algebra, rad L is its radical, and [L, L] is its commutator (derived) subalgebra. All representations of analytic groups are assumed to be analytic and finite dimensional. Received by the editors October 26, 1995 and, in revised form, March 29, 1996. 1991 Mathematics Subject Classification. Primary 22E15, 22E60. ( 1997 American Mathematical Society