$\delta$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions

Abstract

We introduce $\delta$-mock-Novikov algebras as a parameter-dependent mock analogue of Novikov algebras. For $\delta =1$, the commutator of a mock-Novikov algebra defines a Malcev algebra. Every $\delta$-mock-Novikov algebra satisfying ${\mathcal{A}}^2\subseteq \mathrm{Ann}(\mathcal{A})$ is shown to be differentially special. At the operadic level, the binary quadratic operad governing $\delta$-mock-Novikov algebras is quadratically self-dual but not Koszul. We prove that finite-dimensional $\delta$-mock-Novikov algebras are nilpotent in characteristic zero and classify those of dimension at most four. Finally, we study the associated Poisson-type structures and establish relations and constructions among them via polarization, depolarization, and tensor products.

30 pages