Cohomologies, deformations and identities of Lie algebras of vector fields

Right-symmetric algebras satisfy the identity a · (b · c) − (a · b) · c = a · (c · b) − (a · c) · b. Example: the Witt algebra W_{n} (algebra of formal vector fields with n variables) is right-symmetric under multiplication u∂_{i} · v∂_{j} = v∂_{j}(u)∂_{i}. The author developed a cohomology theory for right-symmetric algebras. Connection between right-symmetric and Lie algebra cohomologies was established. It was used in calculation of “hidden” deformations and non-split extensions of four Cartan series of Lie algebras of vector fields. In particular, it is proved that Lie algebras of divergence-free vector fields and Hamiltonian vector fields have new central extensions. Virasoro algebra has one more hidden central extension.

All necessary notions will be introduced in the talk.