Central extensions of infinite-dimensional Lie algebras and Lie groups, Virasoro algebra and generalizations
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.
scientific article; zbMATH DE number 1087253
scientific article