Quantization, deformations, and new homological and categorical methods in mathematical physics

Second cohomology groups for four Cartan series of Lie algebras of vector fields are calculated. We develop right-symmetric cohomology of algebras of vector fields. Some applications of cocyle constructions are given:

• The algebra of divergence-free vector fields on T² has a 5-commutator.
• The Witt algebra W₁ has 4-parametric right-symmetric deformations, but only two of them can be prolongated to global deformation.
• The Witt algebra W_n has a (n²+2n-2)-commutator and it satisfies an identity of degree n²+2n-1.