The Campbell-Hausdorff group and a polar decomposition of graded algebra automorphisms

Alberto Baider and Richard C. Churchill

Vol. 131 (1988), No. 2, 219–235

DOI: 10.2140/pjm.1988.131.219

Abstract

Let A = ∏ _k=k0^∞gr_k(A) be a complete graded (associative or Lie) algebra over a field of characteristic zero, filtered by the decreasing filtration F_j(A) = ∏ _k=j^∞gr_k(A). We let Aut(A) denote the group of filtration preserving automorphisms of A, and Aut₀(A) the subgroup consisting of those elements of Aut(A) which preserve the grading. In this paper we prove that every element of Aut(A) has a unique polar decomposition of the form u₀ exp(d), where u₀ ∈ Aut₀(A) and d : A → A is a filtration increasing derivation.

Mathematical Subject Classification 2000

Primary: 17B70

Secondary: 58F99, 16A03

Milestones

Received: 17 June 1986

Revised: 6 November 1986

Published: 1 February 1988