Mock-Lie bialgebras and mock-Lie analogue of the classical Yang-Baxter equation

TL;DR

This paper introduces mock-Lie bialgebras, explores their relation to the mock-Lie Yang-Baxter equation, and studies solutions using $ ext{O}$-operators, extending classical Lie algebra concepts.

Contribution

It defines mock-Lie bialgebras, introduces the mock-Lie Yang-Baxter equation, and develops methods to construct solutions using $ ext{O}$-operators, providing a new algebraic framework.

Findings

Mock-Lie bialgebras are equivalent to Manin triples of mock-Lie algebras.

Skew-symmetric solutions of the mock-Lie Yang-Baxter equation produce mock-Lie bialgebras.

$ ext{O}$-operators are effective in constructing solutions to the mock-Lie Yang-Baxter equation.

Abstract

The aim of this paper is to introduce the notion of a mock-Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to the introduction the mock-Lie Yang-Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang-Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang-Baxter equation gives a mock-Lie bialgebra. Finally, the notation of $\mathcal O$-operators are studied to construct skew-symmetric solution of mock-Lie Yang-Baxter equation.