The geometric classification of Novikov-Poisson and Gelfand-Dorfman algebras

Abstract

We give the geometric classification of complex 3-dimensional Novikov-Poisson and Gelfand-Dorfman algebras. In particular, we prove that the variety of complex 3-dimensional Novikov-Poisson algebras is 12-dimensional; it has 4 irreducible components and has no rigid algebras; the variety of complex 3-dimensional Gelfand-Dorfman algebras is 11-dimensional; it has 10 irreducible components and has 3 rigid algebras. As an additional result, we have the geometric classification of complex 3-dimensional Novikov algebras: it is 9-dimensional, and it has $4$ irreducible components with one rigid algebra.