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Kupershmidt operators on $3$-BiHom-Poisson color algebras and $3$-BiHom-pre-Poisson color algebras structures

Othmen Ncib, Sergei Silvestrov

Abstract

The purpose of this paper is to introduce the class of noncommutative $3$-BiHom-Poisson color algebras, which is a combination of $3$-BiHom-Lie color algebras and BiHom-associative color algebras under a compatibility condition, called BiHom-Leibniz color identity, and then study their representations and associated Kupershmidt operators. In addition, we introduce the notion of noncommutative $3$-BiHom-pre-Poisson color algebras and investigate the relationship between noncommutative $3$-BiHom-Poisson color algebras and $3$-BiHom-pre-Poisson color algebras via Kupershmidt operators.

Kupershmidt operators on $3$-BiHom-Poisson color algebras and $3$-BiHom-pre-Poisson color algebras structures

Abstract

The purpose of this paper is to introduce the class of noncommutative -BiHom-Poisson color algebras, which is a combination of -BiHom-Lie color algebras and BiHom-associative color algebras under a compatibility condition, called BiHom-Leibniz color identity, and then study their representations and associated Kupershmidt operators. In addition, we introduce the notion of noncommutative -BiHom-pre-Poisson color algebras and investigate the relationship between noncommutative -BiHom-Poisson color algebras and -BiHom-pre-Poisson color algebras via Kupershmidt operators.

Paper Structure

This paper contains 6 sections, 23 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and basic definitions
  3. Noncommutative $3$-BiHom-Poisson color algebras
  4. Noncommutative $3$-BiHom-Poisson color algebras and their representations
  5. Kupershmidt operators on noncommutative $3$-BiHom-Poisson color algebras
  6. Noncommutative $3$-BiHom-pre-Poisson color algebras

Key Result

Proposition 2.2

Let $(\mathfrak{g},\mu,\epsilon)$ be an associative color algebra and $\alpha,\;\beta:\mathfrak{g}\to\mathfrak{g}$ two even commuting linear maps satisfying condition asso-multip-cond, then the tuple $(\mathfrak{g},\mu_{\alpha,\beta},\epsilon,\alpha,\beta)$ is a BiHom-associative color algebra, wher

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Definition 2.3: Bakayoko2
  • Definition 2.4
  • Remark 2.1
  • Proposition 2.2: Yaling-Yan
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.3
  • ...and 66 more