A Poisson-Lie structure on the diffeomorphism group of a circle
\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.
scientific article; zbMATH DE number 716275