Note on faithful representations and a local property of Lie groups

Abstract

Abstract. Let G be any analytic group, let T be a maximal toroid of the radical of G, and let S be a maximal semisimple analytic subgroup of G.If, and Z(L) is the center of L, we show that G has a faithful representation if and (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] ∩ Z(L) ∩ 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.

Kernels of linear representations of Lie groups, locally compact groups, and pro-Lie Groups

“…10.2 Theorem (Nahlus [36]). Let G be a connected Lie group with Lie algebra g := L(G), let r be the solvable radical of the commutator algebra g ′ , and let z be the center of g. Choose a maximal torus T of the solvable radical of G and a maximal semisimple subgroup S. Then G is linear precisely if r ∩ z ∩ L(T) = {0} and S is linear.…”

On the embeddability of certain infinitely divisible probability measures on Lie groups

“…Proof of Corollary 1.4: Let T be the maximal torus in N. Then there exists a representation ρ : G → GL(n, R), for some n ∈ N, such that ker ρ is contained in the center of G and (ker ρ) 0 = T (cf. [16]). Now, [ρ(G), ρ(G)] is an almost algebraic subgroup (cf.…”