Abstract
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. A Novikov–Poisson algebra is a Novikov algebra with a compatible commutative associative algebraic structure, which was introduced to construct the tensor product of two Novikov algebras. In this paper, we commence a study of finite-dimensional Novikov–Poisson algebras. We show the commutative associative operation in a Novikov–Poisson algebra is a compatible global deformation of the associated Novikov algebra. We also discuss how to classify Novikov–Poisson algebras. And as an example, we give the classification of 2-dimensional Novikov–Poisson algebras.
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