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Transposed Novikov-Poisson algebras

Jiarou Jin, Yanyong Hong

TL;DR

The paper defines transposed Novikov-Poisson algebras and establishes their affinization to transposed Poisson algebras via $A[t,t^{-1}]$, connecting the two frameworks. It develops a broad toolkit: tensor-product closure, multiple construction methods (including centroid-based, Kantor-product, and deformations), and a clear link to $\tfrac{1}{2}$-derivations of associated Novikov algebras. It analyzes solvable and simple cases, showing that non-trivial simple transposed NP algebras force the associated Novikov algebra to be simple and that, over algebraically closed fields of characteristic $0$, such algebras are necessarily 1-dimensional. Together, these results classify 2D cases over $\mathbb{C}$, relate transposed NP algebras to transposed Poisson structures, and provide a foundation for constructing and classifying transposed NP algebras in a range of algebraic settings.

Abstract

In this paper, we introduce the definition of transposed Novikov-Poisson algebras, whose affinization are transposed Poisson algebras. Moreover, we show that there is a natural transposed Poisson algebra structure on the tensor product of a transposed Novikov-Poisson algebra and a right differential Novikov-Poisson algebra. A transposed Poisson algebra also naturally arises from a transposed Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. We show that the tensor products of two transposed Novikov-Poisson algebras are also transposed Novikov-Poisson algebras. Several constructions of transposed Novikov-Poisson algebras are presented. Moreover, transposed Novikov-Poisson algebras are closely related to $\frac{1}{2}$-derivations of the associated Novikov algebras. By using $\frac{1}{2}$-derivations, we show that there are non-trivial transposed Novikov-Poisson algebra structures on solvable Novikov algebras with some conditions. We also prove that if a non-trivial transposed Novikov-Poisson algebra is simple, then the associated Novikov algebra is simple. Therefore, if the base field is algebraically closed and of characteristic 0, then any simple transposed Novikov-Poisson is of dimension $1$. Transposed Novikov-Poisson algebra structures on some simple Novikov algebras are also characterized.

Transposed Novikov-Poisson algebras

TL;DR

The paper defines transposed Novikov-Poisson algebras and establishes their affinization to transposed Poisson algebras via , connecting the two frameworks. It develops a broad toolkit: tensor-product closure, multiple construction methods (including centroid-based, Kantor-product, and deformations), and a clear link to -derivations of associated Novikov algebras. It analyzes solvable and simple cases, showing that non-trivial simple transposed NP algebras force the associated Novikov algebra to be simple and that, over algebraically closed fields of characteristic , such algebras are necessarily 1-dimensional. Together, these results classify 2D cases over , relate transposed NP algebras to transposed Poisson structures, and provide a foundation for constructing and classifying transposed NP algebras in a range of algebraic settings.

Abstract

In this paper, we introduce the definition of transposed Novikov-Poisson algebras, whose affinization are transposed Poisson algebras. Moreover, we show that there is a natural transposed Poisson algebra structure on the tensor product of a transposed Novikov-Poisson algebra and a right differential Novikov-Poisson algebra. A transposed Poisson algebra also naturally arises from a transposed Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. We show that the tensor products of two transposed Novikov-Poisson algebras are also transposed Novikov-Poisson algebras. Several constructions of transposed Novikov-Poisson algebras are presented. Moreover, transposed Novikov-Poisson algebras are closely related to -derivations of the associated Novikov algebras. By using -derivations, we show that there are non-trivial transposed Novikov-Poisson algebra structures on solvable Novikov algebras with some conditions. We also prove that if a non-trivial transposed Novikov-Poisson algebra is simple, then the associated Novikov algebra is simple. Therefore, if the base field is algebraically closed and of characteristic 0, then any simple transposed Novikov-Poisson is of dimension . Transposed Novikov-Poisson algebra structures on some simple Novikov algebras are also characterized.
Paper Structure (15 sections, 40 theorems, 126 equations, 1 table)

This paper contains 15 sections, 40 theorems, 126 equations, 1 table.

Table of Contents

  1. Introduction
  2. Properties of transposed Novikov-Poisson algebras and the relationships with transposed Poisson algebras
  3. Transposed Novikov-Poisson algebras and transposed Poisson algebras
  4. Properties of transposed Novikov-Poisson algebras
  5. A construction of transposed Novikov-Poisson algebras from commutative associative algebras
  6. Independence of Novikov-Poisson algebras and transposed Novikov-Poisson algebras
  7. Identities in transposed Novikov-Poisson algebras and classification of 2-dimensional transposed Novikov-Poisson algebras over $\mathbb{C}$
  8. Tensor products
  9. Constructions from known transposed Novikov-Poisson algebras
  10. $\frac{1}{2}$-derivations of Novikov algebras and transposed Novikov-Poisson algebra structures
  11. $\frac{1}{2}$-derivations of Novikov algebras
  12. $\frac{1}{2}$-derivations of solvable Novikov algebras
  13. Simple transposed Novikov-Poisson algebras
  14. On simple transposed Novikov-Poisson algebras
  15. Transposed Novikov-Poisson algebras with simple Novikov 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 on $A[t,t^{-1}]\coloneqq A\otimes {\bf k}[t,t^{-1}]$ by for all $x,y\in A$ and $m, n\in\mathbb{Z}$, where $x t^m:=x\otimes t^{m}$. Then $(A[t,t^{-1}],[\cdot,\cdot])$ is a Lie algebra if and only if $(A, \circ)$ is a Novikov algebra.

Theorems & Definitions (102)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 92 more