The algebraic and geometric classification of $\delta$-Novikov algebras

Abstract

The notion of $\delta$-Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like $\delta$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a $2$-dimensional simple algebra for $\delta=-1.$ The present paper is dedicated to the study of $3$-dimensional $\delta$-Novikov algebras for $\delta \notin \big\{0,1\big\}.$ The algebraic and geometric classifications of complex $3$-dimensional $\delta$-Novikov algebras are given. As a corollary, we prove that there are no simple $3$-dimensional $\delta$-Novikov algebras.