The Campbell-Hausdorff group and a polar decomposition of graded algebra automorphisms
type: Item
instance of: scholarly article
title: The Campbell-Hausdorff group and a polar decomposition of graded algebra automorphisms
zbMATH Open document ID: 0599.17002
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(A=\prod^{\infty}_{k=k_ 0}gr_ k(A)\) be a complete (not necessarily associative) algebra over a field of characteristic zero, filtered by the decreasing filtration \(F_ j(A)=\prod^{\infty}_{k=j}gr_ k(A)\). Let Aut(A) (resp. \(Aut_ 0(A))\) be the group of filtration (resp. gradation) preserving automorphisms of A. \(Aut_ 0(A)\subseteq Aut(A)\), and the grading insures there is a (split) surjective homomorphism \(g_ r: Aut(A)\to Aut_ 0(A)\). We prove that the group sequence \[ e\quad \to \quad F_ 1(Der(A))\to^{\exp}Aut(A)\to^{gr}Aut_ 0(A)\quad \to \quad e \] is exact. (Here Der(A) is the Lie algebra of filtration preserving derivations of A and \(F_ 1(Der(A))\) is the so-called Campbell-Hausdorff group of Der(A). The group operation in Der(A) is defined by the usual Campbell-Hausdorff formula.) Thus every automorphism in Aut(A) admits a unique polar decomposition \(u_ 0 \exp (d)\), where \(u_ 0\in Aut_ 0(A)\) and d is a derivation satisfying \(d(F_ j(A))\subseteq F_{j+1}(A)\) for all j. As we show in the applications, the result generalizes the well known fact that every formal diffeomorphism \(\phi\) of \(({\mathbb{R}}^ N,0)\) is the product of a linear map (\(\phi\) '(0)) with the time-one map of the flow of a unique (formal) vector field vanishing to order two at 0.
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17A36
zbMATH DE Number: 3966348
zbMATH Keywords: exact group sequence
zbMATH Keywords: filtered algebras
zbMATH Keywords: polar decomposition of graded algebra automorphisms
zbMATH Keywords: Campbell-Hausdorff group
author: Alberto Baider
author: Richard C. Churchill
MaRDI profile type: Publication
full work available at URL: pjm.1988.131.219
OpenAlex ID: W1976959832
DOI: 10.2140/PJM.1988.131.219
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description: scientific article; zbMATH DE number 3966348