$\delta$-Novikov and $\delta$-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 $\delta$. Unlike Novikov algebras, they have simple non-commutative finite-dimensional algebras for $\delta =-1$; on the other hand, for every $\delta \ne 1$, not all of them embed into commutative associative algebras with a $\delta$-derivation. We introduce $\delta$-Novikov--Poisson algebras, extending several theorems from classical Novikov--Poisson algebras. For $\delta \ne 1$, the commutator algebras of $\delta$-Novikov algebras are metabelian Lie algebras, and every metabelian Lie algebra can be embedded into a suitable $\delta$-Novikov algebra with respect to the commutator product. We construct $\delta$-Poisson and transposed $\delta$-Poisson algebras through $\delta$-derivations on commutative associative algebras, and show that for $\delta \ne 1$ not every transposed $(\delta +1)$-Poisson algebra comes from a $\delta$-Novikov--Poisson algebra. Finally, we prove that the $\delta$-Novikov operad is not Koszul for any value of $\delta$, extending the known results for $\delta =1$ and $\delta =0$, and that a minimal ideal $I$ of a $\delta$-Novikov algebra is simple or satisfies $I^2=0$.

arXiv commentThe article was accepted for publication in the Bulletin of the Iranian Mathematical Society but subsequently rejected four months later on the grounds that it was already available on arXiv.org