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arXiv · 2505.08043

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

Ivan Kaygorodov
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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 $(δ=0)$.

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BibTeXRIS

Ivan Kaygorodov. 2025-05-12. $δ$-Novikov and $δ$-Novikov--Poisson algebras. https://arxiv.org/abs/2505.08043

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