$\delta$-Novikov and $\delta$-Novikov--Poisson algebras

TL;DR

This paper explores the structure and properties of $oldsymbol{ ext{}oldsymbol{ extdelta} ext{-Novikov} ext{}}$ and $oldsymbol{ ext{}oldsymbol{ extdelta} ext{-Novikov--Poisson} ext{}}$ algebras, extending classical theories and analyzing their operads, revealing new algebraic behaviors and embedding properties.

Contribution

It introduces $oldsymbol{ extdelta}$-Novikov--Poisson algebras, studies their structure, and proves the non-Koszul property of their operads for all $oldsymbol{ extdelta}$ values.

Findings

$oldsymbol{ extdelta}$-Novikov algebras have non-commutative simple finite-dimensional cases for $oldsymbol{ extdelta=-1}$.

When $oldsymbol{ extdelta eq 1}$, $oldsymbol{ extdelta}$-Novikov algebras are metabelian Lie-admissible.

The operad for $oldsymbol{ extdelta}$-Novikov algebras is not Koszul for any $oldsymbol{ extdelta}$.

Abstract

This article considers the structure and properties of $\delta$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $\delta$. It looks like $\delta$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have non-commutative simple finite-dimensional algebras for $\delta=-1.$ Additionally, we introduce $\delta$-Novikov--Poisson algebras, extending several theorems from the classical Novikov--Poisson algebras. Specifically, we consider the commutator structure $[a, b] = a \circ b - b \circ a$ of $\delta$-Novikov algebras, proving that when $\delta \neq 1$, these algebras are metabelian Lie-admissible. Moreover, we prove that every metabelian Lie algebra can be embedded into a suitable $\delta$-Novikov algebra with respect to the commutator product. We further consider the construction of $\delta$-Poisson and transposed $\delta$-Poisson algebras through $\delta$-derivations on the commutative associative algebras. Finally, we analyze the operad associated with the variety of $\delta$-Novikov algebras, proving that it is not Koszul for any value of $\delta$. This result extends known results for the Novikov operad $(\delta=1)$ and the bicommutative operad $(\delta=0)$.