$δ$-Novikov and $δ$-Novikov--Poisson algebras

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 $(\delta=0)$.