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

TL;DR

This paper classifies 3-dimensional complex $oldsymbol{ extit{ extdelta}}$-Novikov algebras algebraically and geometrically, revealing their structure and proving the non-existence of simple 3-dimensional $oldsymbol{ extit{ extdelta}}$-Novikov algebras for certain parameters.

Contribution

It provides the first comprehensive algebraic and geometric classification of 3-dimensional $ extdelta$-Novikov algebras for $ extdelta otin igrace{0,1}$, including the proof of no simple cases.

Findings

Classified all 3D $ extdelta$-Novikov algebras algebraically and geometrically.

Proved that no simple 3D $ extdelta$-Novikov algebras exist for $ extdelta otin igrace{0,1}$.

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.