The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence

Vladimir V. Bavula

doi:10.1016/j.crma.2013.12.001

Algebra/Lie algebras

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence
[Le groupe dʼautomorphismes de lʼalgèbre de Lie des champs de vecteurs formellement analytiques à divergence constante]

Présenté par : the Editorial Board

Vladimir V. Bavula ¹

¹ Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 85-88.

Résumé
Abstract

Soit $S_{n} = K [[x_{1}, \dots, x_{n}]]$ lʼalgèbre des séries formelles sur un corps K de caractéristique zéro, $S_{n}^{c}$ le groupe des automorphismes continus de $S_{n}$ de jacobien constant et ${Div}_{n}^{c}$ lʼalgèbre de Lie des dérivations de $S_{n}$ à divergence constante. Nous montrons les identités ${Aut}_{Lie} ({Div}_{n}^{c}) = {Aut}_{Lie, c} ({Div}_{n}^{c}) ≃ S_{n}^{c}$ .

Let $S_{n} = K [[x_{1}, \dots, x_{n}]]$ be the algebra of power series over a field K of characteristic zero, $S_{n}^{c}$ be the group of continuous automorphisms of $S_{n}$ with constant Jacobian, and ${Div}_{n}^{c}$ be the Lie algebra of derivations of $S_{n}$ with constant divergence. We prove that ${Aut}_{Lie} ({Div}_{n}^{c}) = {Aut}_{Lie, c} ({Div}_{n}^{c}) ≃ S_{n}^{c}$ .

Reçu le : 2013-11-10
Accepté le : 2013-12-03
Publié le : 2014-01-04

DOI : 10.1016/j.crma.2013.12.001

Affiliations des auteurs :

Vladimir V. Bavula ¹

¹ Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

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     title = {The groups of automorphisms of the {Lie} algebras of formally analytic vector fields with constant divergence},
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Vladimir V. Bavula. The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 85-88. doi : 10.1016/j.crma.2013.12.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.001/

Bibliographie
Cité par

[1] V.V. Bavula The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence | arXiv

[2] A.N. Rudakov Automorphism groups of infinite-dimensional simple Lie algebras, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 33 (1969), pp. 748-764

[3] A.N. Rudakov Subalgebras and automorphisms of Lie algebras of Cartan type, Funkc. Anal. Prilozh., Volume 20 (1986) no. 1, pp. 83-84

Alexei Kanel-Belov; Jie-Tai Yu; Andrey Elishev On the augmentation topology of automorphism groups of affine spaces and algebras, International Journal of Algebra and Computation, Volume 28 (2018) no. 08, p. 1449 | DOI:10.1142/s0218196718400040

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