Jacobi algebras and Jacobi Novikov-Poisson algebras

Abstract

In this paper, we introduce the notion of Jacobi Novikov-Poisson algebras and demonstrate that their affinization yields Jacobi algebras. We note that every unital differential Novikov-Poisson algebra is also a Jacobi Novikov-Poisson algebra. Additionally, any Jacobi Novikov-Poisson algebra gives rise to a Jacobi algebra, either by taking the commutator bracket of its underlying Novikov algebra or by using a derivation. We provide classifications of low-dimensional Jacobi Novikov-Poisson algebras including those of dimensions 2 and 3 over $\mathbb{C}$ up to isomorphism and show that the tensor product of two such algebras remains a Jacobi Novikov-Poisson algebra. Several further constructions of Jacobi Novikov-Poisson algebras from existing ones are also presented. The notion of Frobenius Jacobi Novikov-Poisson algebras is introduced, and several equivalent characterizations are established in terms of quadratic structures and integrals. Classifications of quadratic Jacobi Novikov-Poisson algebras of dimensions 2 and 3 over $\mathbb{C}$ are given. Finally, we provide an explicit construction of Frobenius Jacobi algebras using finite-dimensional quadratic Jacobi Novikov-Poisson algebras and finite-dimensional quadratic right Jacobi Novikov-Poisson algebras.