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On the transposed Poisson n-Lie algebras

Farukh Mashurov

TL;DR

The paper investigates transposed Poisson $n$-Lie structures arising from unital commutative associative algebras, proving that a natural compatibility yields a strong transposed Poisson $n$-Lie algebra and providing examples such as the Dzhumadil'daev construction. It then generalizes a simplicity criterion, showing that a transposed Poisson $n$-Lie algebra is simple if and only if its associated $n$-Lie algebra is simple, and applies this to simple linearly compact cases. In addition, the authors demonstrate that the strong condition holds in the unital setting for a broad class of algebras but fails for the free transposed Poisson $3$-Lie algebra, highlighting limits of the strong condition for $n\ge 3$. Overall, the results connect transposed Poisson structures with classical $n$-Nambu–Poisson-like algebras and contribute to classification and construction of simple transposed Poisson $n$-Lie algebras.

Abstract

We study unital commutative associative algebras and their associated n-Lie algebras, showing that they are strong transposed Poisson n-Lie algebras under specific compatibility conditions. Furthermore, we generalize the simplicity criterion for transposed Poisson algebras, proving that a transposed Poisson n-Lie algebra is simple if and only if its associated n-Lie algebra is simple. In addition, we study the strong condition for transposed Poisson n-Lie algebras, proving that it fails in the case of a free transposed Poisson 3-Lie algebra.

On the transposed Poisson n-Lie algebras

TL;DR

The paper investigates transposed Poisson -Lie structures arising from unital commutative associative algebras, proving that a natural compatibility yields a strong transposed Poisson -Lie algebra and providing examples such as the Dzhumadil'daev construction. It then generalizes a simplicity criterion, showing that a transposed Poisson -Lie algebra is simple if and only if its associated -Lie algebra is simple, and applies this to simple linearly compact cases. In addition, the authors demonstrate that the strong condition holds in the unital setting for a broad class of algebras but fails for the free transposed Poisson -Lie algebra, highlighting limits of the strong condition for . Overall, the results connect transposed Poisson structures with classical -Nambu–Poisson-like algebras and contribute to classification and construction of simple transposed Poisson -Lie algebras.

Abstract

We study unital commutative associative algebras and their associated n-Lie algebras, showing that they are strong transposed Poisson n-Lie algebras under specific compatibility conditions. Furthermore, we generalize the simplicity criterion for transposed Poisson algebras, proving that a transposed Poisson n-Lie algebra is simple if and only if its associated n-Lie algebra is simple. In addition, we study the strong condition for transposed Poisson n-Lie algebras, proving that it fails in the case of a free transposed Poisson 3-Lie algebra.
Paper Structure (5 sections, 10 theorems, 19 equations)

This paper contains 5 sections, 10 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unital transposed Poisson $n$-Lie algebras
  4. Simplicity of transposed Poisson $n$-Lie algebras
  5. A strong transposed Poisson $3$-Lie algebras

Key Result

Proposition 2.5

The following identities hold true in a transposed Poisson $n$-Lie algebra $(A,\cdot, [\cdot,\ldots,\cdot]):$ where $h,x_i,y_i\in A$ and $i=1,\ldots,n+1.$

Theorems & Definitions (17)

  • Conjecture 1.1
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • ...and 7 more