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Jacobi algebras and Jacobi Novikov-Poisson algebras

Chengyang Lu, Yanyong Hong

TL;DR

The paper defines Jacobi Novikov-Poisson algebras as a bridge between Novikov and Jacobi structures and proves that affinization on $A[t,t^{-1}]$ yields a Jacobi algebra precisely when $A$ is Jacobi NP. It shows that unital differential Novikov-Poisson algebras are Jacobi NP and that Jacobi NP algebras produce Jacobi algebras via commutator or derivation, with tensor-product constructions preserving Jacobi structure. It provides complete low-dimensional classifications in dimensions 2 and 3 over $\mathbb{C}$, develops a Frobenius theory via integrals and quadratic forms, and gives explicit constructions of Frobenius Jacobi algebras from quadratic Jacobi NP algebras and their right counterparts, including concrete examples. Together, these results extend the algebraic toolkit for Jacobi-type structures and illuminate connections to Poisson geometry through affinization, duality, and Frobenius theory, with potential implications for representations and deformation theory.

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.

Jacobi algebras and Jacobi Novikov-Poisson algebras

TL;DR

The paper defines Jacobi Novikov-Poisson algebras as a bridge between Novikov and Jacobi structures and proves that affinization on yields a Jacobi algebra precisely when is Jacobi NP. It shows that unital differential Novikov-Poisson algebras are Jacobi NP and that Jacobi NP algebras produce Jacobi algebras via commutator or derivation, with tensor-product constructions preserving Jacobi structure. It provides complete low-dimensional classifications in dimensions 2 and 3 over , develops a Frobenius theory via integrals and quadratic forms, and gives explicit constructions of Frobenius Jacobi algebras from quadratic Jacobi NP algebras and their right counterparts, including concrete examples. Together, these results extend the algebraic toolkit for Jacobi-type structures and illuminate connections to Poisson geometry through affinization, duality, and Frobenius theory, with potential implications for representations and deformation theory.

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 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 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.
Paper Structure (8 sections, 20 theorems, 69 equations)

This paper contains 8 sections, 20 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Basic results of Jacobi Novikov-Poisson algebras and the relationships with Jacobi algebras
  3. The relationships between Jacobi Novikov-Poisson algebras and Jacobi algebras
  4. Classifications of Jacobi Novikov-Poisson algebras in low diemnsions
  5. Constructions of Jacobi Novikov-Poisson algebras from known Jacobi Novikov-Poisson algebras
  6. Frobenius Jacobi Novikov-Poisson algebras and Frobenius Jacobi algebras
  7. Modules, integrals and Frobenius Jacobi Novikov-Poisson algebras
  8. Classifications of quadratic Jacobi Novikov-Poisson algebras in low dimensions and a construction of Frobenius Jacobi algebras

Key Result

Theorem 1.1

BN Let k be a field of characteristic 0 and $A$ be a vector space over k endowed with a binary operation $\circ$. Define a binary operation $[\cdot,\cdot]$ on $A[t,t^{-1}]\coloneqq A\otimes {\bf k}[t,t^{-1}]$ by where $at^m:=a\otimes t^{m}$. Then $(A[t,t^{-1}],[\cdot,\cdot])$ is a Lie algebra if and only if $(A, \circ)$ is a Novikov algebra.

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 44 more