Central extensions of infinite-dimensional Lie algebras and Lie groups, Virasoro algebra and generalizations
type: Item
instance of: scholarly article
title: Central extensions of infinite-dimensional Lie algebras and Lie groups, Virasoro algebra and generalizations
zbMATH Open document ID: 0892.17018
DOI: 10.1016/0034-4877(96)89288-3
author: Claude Roger
published in: Reports on Mathematical Physics
publication date: 1998-01-12T00:00:00Z
review text: This is a survey article on central extensions of Lie algebras of vector fields and of the corresponding groups of diffeomorphisms. The author first discusses generalities about central extensions of Lie algebras and Lie groups and the relation to cohomology and homology. Next, he discusses the relations between central extensions, projective representations and coadjoint orbits and their symplectic structure. A relation between the cohomologies with coefficients in the adjoint representation and the cohomology with trivial coefficients is described. Then the author discusses the fundamental example of the Virasoro algebra and the Virasoro group. Next, he describes central extensions of current algebras and Kac-Moody algebras and their relation to cyclic homology and cohomology. The next algebras treated are the Lie algebras of differential operators and of pseudo-differential operators on the circle, where central extensions are described using the noncommutative residue and the relation to central extensions of the subalgebra of vector fields is discussed. Then various generalizations of the Virasoro algebra (algebras of meromorphic vector fields on Riemann surfaces and superalgebraic versions of the Virasoro algebra) and their central extensions are described. Next, the author discusses the algebras of (globally) Hamiltonian vector fields on a symplectic manifold, of unimodular vector fields on a manifold equipped with a volume form, and of contact vector fields on a contact manifold. In the unimodular case, applications to equations from hydrodynamics and extensions of the associated diffeomorphism groups are also treated. Finally, a unified approach to various cases of central extensions via infinite linear groups is described.
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 1087253
zbMATH Keywords: Lie algebra of vector fields
zbMATH Keywords: central extension
zbMATH Keywords: survey
zbMATH Keywords: groups of diffeomorphisms
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: Virasoro group
zbMATH Keywords: current algebras
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: cyclic homology
zbMATH Keywords: cohomology
zbMATH Keywords: unimodular vector fields
zbMATH Keywords: contact vector fields
zbMATH Keywords: algebras of meromorphic vector fields
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Hamiltonian vector fields
reviewed by: Andreas Cap
MaRDI profile type: Publication
full work available at URL: 0034-4877(96)89288-3
OpenAlex ID: W2065619479
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description: scientific article; zbMATH DE number 1087253