$δ$-Novikov and $δ$-Novikov--Poisson algebras
Ivan Kaygorodov
TL;DR
This work introduces $\\delta$-Novikov algebras as a parametric generalization of Novikov algebras and develops the related $\\delta$-Novikov--Poisson framework. It analyzes the commutator structure, showing that for $\\delta \\neq 1$ the commutator yields metabelian Lie-admissible algebras and that every metabelian Lie algebra embeds into a suitable $\\delta$-Novikov algebra; it also provides $\\\\\\\\\\\\\\\\\\\\\ ext{derivation}$-based constructions yielding $\\delta$-Poisson, transposed $\\delta$-Poisson, and $\\delta$-Gelfand–Dorfman algebras. The operadic investigation proves the $\\\\\\\\\\\\\\\\\\\\\ ext{delta}$-Novikov operad is not Koszul for any $\\delta$, extending known results for $\\delta=1$ and $\\delta=0$. Together with solvability/nilpotency analogues, these results broaden the landscape of Novikov-type algebras and connect them to Poisson-type and GD structures via $\\delta$-derivations.
Abstract
This article considers the structure and properties of $δ$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $δ$. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have non-commutative simple finite-dimensional algebras for $δ=-1.$ Additionally, we introduce $δ$-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 $δ$-Novikov algebras, proving that when $δ\neq 1$, these algebras are metabelian Lie-admissible. Moreover, we prove that every metabelian Lie algebra can be embedded into a suitable $δ$-Novikov algebra with respect to the commutator product. We further consider the construction of $δ$-Poisson and transposed $δ$-Poisson algebras through $δ$-derivations on the commutative associative algebras. Finally, we analyze the operad associated with the variety of $δ$-Novikov algebras, proving that it is not Koszul for any value of $δ$. This result extends known results for the Novikov operad $(δ=1)$ and the bicommutative operad $(δ=0)$.