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Twisted Rota-Baxter family operators on Hom-associative algebras

Wen Teng, Yunpeng Xiao

Abstract

In this paper, we first define twisted Rota-Baxter family operators on Hom-associative algebras indexed by a semigroup $Ω$. Then we introduce and study Hom-NS-family algebras as the underlying structures of twisted Rota-Baxter family operators. Meanwhile, We show that a Hom-NS-family algebra induces an ordinary Hom-NS-algebra on the tensor product with the semigroup algebra. Moreover, we define the cohomology of a twisted Rota-Baxter family operator. This cohomology can also be viewed as the cohomology of a certain Hom-$Ω$-associative algebra with coefficients in a suitable bimodule. Finally, we examine deformations of twisted Rota-Baxter family operators and demonstrate that they are governed by the aforementioned cohomology. The concept of Nijenhuis elements linked to a twisted Rota-Baxter family operator is introduced to provide a sufficient condition for its rigidity.

Twisted Rota-Baxter family operators on Hom-associative algebras

Abstract

In this paper, we first define twisted Rota-Baxter family operators on Hom-associative algebras indexed by a semigroup . Then we introduce and study Hom-NS-family algebras as the underlying structures of twisted Rota-Baxter family operators. Meanwhile, We show that a Hom-NS-family algebra induces an ordinary Hom-NS-algebra on the tensor product with the semigroup algebra. Moreover, we define the cohomology of a twisted Rota-Baxter family operator. This cohomology can also be viewed as the cohomology of a certain Hom--associative algebra with coefficients in a suitable bimodule. Finally, we examine deformations of twisted Rota-Baxter family operators and demonstrate that they are governed by the aforementioned cohomology. The concept of Nijenhuis elements linked to a twisted Rota-Baxter family operator is introduced to provide a sufficient condition for its rigidity.

Paper Structure

This paper contains 5 sections, 17 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Twisted Rota-Baxter family operators on Hom-associative algebras
  3. Hom-NS-family algebras
  4. Cohomology of twisted Rota-Baxter family operators
  5. Deformations of twisted Rota-Baxter family operators

Key Result

Proposition 2.9

Let $\{R_\alpha:V\rightarrow L\}_{\alpha\in\Omega}$ be an $\Phi$-twisted Rota-Baxter family operator. Then the map is an $\bar{\Phi}$-twisted Rota-Baxter operator on $V\otimes \mathbb{K}\Omega$ over the Hom-associative algebra $(L\otimes \mathbb{K}\Omega,\bar{\cdot},\bar{p})$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Proposition 2.9
  • proof
  • ...and 42 more