Cohomologies and abelian extensions of Novikov algebras (Q6596132)

Novikov algebras appeared in the study of Hamiltonian operators in the formal calculus of variations. They also played a crucial role for the classification of linear Poisson brackets of hydrodynamical type. Note that Novikov algebras are special left-symmetric algebras (or pre-Lie algebras). In this paper, the authors present a cohomology of Novikov algebras (in fact, this cohomology is a cohomology of left-symmetric algebras), which is a slight modification of the cohomology of right-symmetric algebras given by A. Dzhumadil'daev. By applying this cohomology of Novikov algebras, this paper shows that all abelian extensions of a Novikov algebra by a module are classified by the subspace of the second cohomology group given by quasi-associative bilinear maps.