Nijenhuis operators and mock-Lie bialgebras

TL;DR

This paper extends the theory of Nijenhuis mock-Lie algebras to bialgebras, exploring their structure, deformations, extensions, and solutions to a mock-Lie Yang-Baxter equation, thus broadening the algebraic framework.

Contribution

It generalizes Nijenhuis mock-Lie algebra results to bialgebras, introduces cohomology for deformations, and defines mock-Lie-Yang-Baxter equations and $ ext{O}$-operators.

Findings

Characterization of Nijenhuis mock-Lie bialgebras via matched pairs and Manin triples.

Development of deformation theory and cohomology for Nijenhuis mock-Lie algebras.

Introduction of mock-Lie-Yang-Baxter equation and solutions leading to bialgebras.

Abstract

A Nijenhuis mock-Lie algebra is a mock-Lie algebra equipped with a Nijenhuis operator. The purpose of this paper is to extend the well-known results about Nijenhuis mock-Lie algebras to the realm of mock-Lie bialgebras. It aims to characterize Nijenhuis mock-Lie bialgebras by generalizing the concepts of matched pairs and Manin triples of mock-Lie algebras to the context of Nijenhuis mock-Lie algebras. Moreover, we discuss formal deformation theory and explore infinitesimal formal deformations of Nijenhuis mock-Lie algebras, demonstrating that the associated cohomology corresponds to a deformation cohomology. Moreover, we define abelian extensions of Nijenhuis mock-Lie algebras and show that equivalence classes of such extensions are linked to cohomology groups. The coboundary case leads to the introduction of an admissible mock-Lie-Yang-Baxter equation (mLYBe) in Nijenhuis mock-Lie algebras, for which the antisymmetric solutions give rise to Nijenhuis mock-Lie bialgebras. Furthermore, the notion of $\mathcal O$-operator on Nijenhuis mock-Lie algebras is introduced and connected to mock-Lie-Yang-Baxter equation.