A Lie algebraic approach to Novikov algebras

Abstract

Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra. Thus it is useful to relate the study of Novikov algebras to the theory of Lie algebras. In this paper, we will try to realize Novikov algebras through a Lie algebraic approach. Such a realization could be important in physics and geometry. We find that all transitive Novikov algebras in dimension ≤3 can be realized as the Novikov algebras obtained through Lie algebras and their compatible linear (global) deformations.