δ-Novikov and δ-Novikov--Poisson algebras

2025/05/12 by Kaygorodov, Ivan
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2505.08043

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 ∘ b - b ∘ a of δ-Novikov algebras, proving that when δ≠ 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).

Citations

• A characterisation of Lie algebras using ideals and subalgebras
• Polynomial identities in Novikov algebras
• Anti-associative algebras
• Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras
• Polynomial identities of bicommutative algebras, Lie and Jordan elements
• Novikov algebras and Novikov structures on Lie algebras
• Koszul duality for operads

Related

• Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras
• Hom-Novikov Algebras and Hom-Novikov-Poisson Algebras
• A twisted generalization of Novikov-Poisson algebras
• Differential Novikov algebras
• Lie and pre-Lie theory of Novikov algebras