A Poisson-Lie structure on the diffeomorphism group of a circle
type: Item
instance of: scholarly article
title: A Poisson-Lie structure on the diffeomorphism group of a circle
zbMATH Open document ID: 0866.17018
DOI: 10.1007/BF00761141
author: J. Grabowski
published in: Letters in Mathematical Physics
publication date: 1995-01-31T00:00:00Z
review text: \textit{M. E. Shanks} and \textit{L. E. Pursell} proved [``The Lie algebra of a smooth manifold'', Proc. Am. Math. Soc. 5, 468-472 (1954; Zbl 0055.42105)] that maximal ideals of the Lie algebra \({\mathcal H}(M)\) of all smooth vector fields on a compact manifold \(M\) consist of those vector fields which are flat at a given point of \(M\). Several authors have studied ideals of different type Lie algebras of vector fields, for instance, maximal ideals of the Lie algebra \({\mathcal H}(M)\) for non-compact \(M\) were described by \textit{J. Vanžura} [``Maximal ideals in the Lie algebra of vector fields'', Commentat. Math. Univ. Carol. 29, 267-278 (1988; Zbl 0702.57019)] in terms of the Stone-Čech compactification of \(M\). The algebraic approach developed by \textit{J. Grabowski} [``Isomorphisms and ideals of the Lie algebras of vector fields'', Invent. Math. 50, 13-33 (1978; Zbl 0387.57009) and ``The Lie algebras of vector fields on manifolds'', (in Polish), Thesis, Warsaw Univ. (1981)] gives the description for a larger class of the Lie algebras of vector fields, e.g. the vector fields tangent to a given foliation and for analytic cases as well. All the maximal ideals turn out to be simultaneously modules over the rings of the corresponding class of functions. It was conjectured by Vanžura in the early eighties that it is in fact true for any ideal of the Lie algebra \({\mathcal H}(M)\). This note contains a result describing all ideals of a larger class of the Lie algebras of vector fields (including analytic cases) which easily implies Vanžura's conjecture.
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 57R99
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 716275
zbMATH Keywords: Lie algebras of vector fields
zbMATH Keywords: Vanžura's conjecture
MaRDI profile type: Publication
Wikidata QID: Q115394899
cites work: Q3488165
cites work: Q3346993
cites work: Kähler geometry of the infinite-dimensional homogeneous space \(M=Diff_+(S^ 1)/Rot(S^ 1)\)
cites work: Coadjoint orbits of the Virasoro group
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description: scientific article; zbMATH DE number 716275