Some skew linear groups associated with Lie algebras of characteristic zero

Abstract

Let F be a field, L a Lie F-algebra of finite dimension d, and U the universal enveloping algebra of L. Then U is a Noetherian domain [3, p. 1661. Let D denote its division ring of quotients. Suppose charF= 0. Lichtman in [S, 63 considers some group theoretic properties of GL(n, D) and its subgroups. In this note we pursue this investigation a little further. For fuller statements of both Lichtman’s results and our own see Section 2 below. Here we merely summarize a few of the more interesting points. Suppose G is a finitely generated subgroup of GL(n, D). We show for all but a finite number of primes q that G has a normal subgroup of finite index that is residually a finite q-group. In particular of course G is residually finite. Further it follows that G is torsion-free by finite and hence if G is periodic then G is finite. Thus [S] a periodic subgroup P of GL(n, D) is locally finite. We also show that such a P is isomorphic to a subgroup of GL(n, C) and hence that [S] the group P has an abelian normal subgroup with index finite and bounded in terms of n only. Suppose G, is a subgroup of GL(n, D) containing no free subgroup of rank two. Then the structure of Go is described by Theorem A of [6]. We give an alternative approach, yielding some improvement of the bounds involved. As a consequence we produce a sharper bound for the derived length of a soluble subgroup of GL(n, D). Assume again that G is a finitely generated subgroup of GL(n, D). An easy result is that if every finite image of G is soluble, then G is soluble. In order to proceed much further with elucidating the structure of G along the lines suggested by the theory of linear groups, we would need something like the nilpotence of G, given that the finite images of G are nilpotent. Unfortunately the unipotent elements present problems. This is frequently the case where imperfect ground fields are involved. (To see that is the situation here refer to Section 1 below.) We have only been able to prove, under considerably weaker hypotheses that G is nilpotent modulo 155 0021-8693/89 93.00