On faithful representations of the holomorph of Lie groups

Abstract

Let G be a Lie group and let Aut(G) denote the group of all automorphisms of the Lie group G, endowed with its natural structure of a Lie group. The natural semidirect product G >~ Aut(G) is called the holomorph of G. In [2], Hochschild has shown that if the nilradical (that is, the maximum nilpotent analytic normal subgroup) N of a real or complex analytic group G is simply connected, then the holomorph of G is faithfully representable. Moreover, one of the authors in [4] has given an intrinsic characterization of those complex analytic groups whose holomorph is faithfully representable. In this paper, we are interested in determining when the holomorph of a Lie group G is faithfully representable (that is, admits a faithful finite-dimensional continuous representation). Our first result (Theorem 1) answers this question when G is an f.c.c. Lie group. Also in this work, we extend the result of [4] to real analytic groups (Theorem 2). Our method is entirely different from that of [4] and may be adapted to cover the complex case as well. We would like to thank the referee for making valuable suggestions for improvements of this paper.