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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have