On unitary automorphisms of solvable Lie algebras

Oldřich Kowalski

Vol. 82 (1979), No. 1, 133–143

DOI: 10.2140/pjm.1979.82.133

Abstract

Let V be a finite dimensional vector space over real numbers. An automorphism A of V is called unitary if it is semisimple and all its eigenvalues are complex units. Particularly, all periodic automorphisms, i.e., such that T^k = identity for some integer k, are unitary. The aim of this paper is to prove the following Theorem. Let g be an n-dimensional real Lie algebra admitting a unitary automorphism without nonzero fixed vectors. Then g admits a periodic automorphism without nonzero fixed vectors and of order k, where k ≤ 5^n∕4 for n even, and k ≦ 2.5^(n−1)∕4 for n odd.

Mathematical Subject Classification 2000

Primary: 17B40

Milestones

Received: 14 August 1975

Revised: 21 December 1978

Published: 1 May 1979