Deformations of some infinite-dimensional Lie algebras

Abstract

The concept of a versal deformation of a Lie algebra is investigated and obstructions to extending an infinitesimal deformation to a higher-order one are described. The rigidity of the Witt algebra and the Virasoro algebra is deduced from cohomology computations for certain Lie algebras of vector fields on the real line. The Lie algebra of vector fields on the line that vanish at the origin also turns out to be rigid. All the affine Lie algebras are rigid; this is derived from the cohomology of their maximal nilpotent subalgebra. On the other hand, the maximal nilpotent subalgebras in both the Virasoro and affine cases are not rigid and have interesting nontrivial deformations (in fact, most vector field Lie algebras are not rigid).