The Campbell-Hausdorff group and a polar decomposition of graded algebra automorphisms
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.
polar decomposition of graded algebra automorphisms
scientific article; zbMATH DE number 3966348