Manin triples, bialgebras and Yang-Baxter equation of $A_3$-associative algebras

TL;DR

The work extends bialgebra theory to $A_3$-associative algebras, a Lie-admissible generalization of associative structures, by formulating Manin triples, bialgebras, and a Yang-Baxter framework in this setting. It develops representations and matched pairs, introduces quadratic forms and the double construction, and shows the equivalence between Manin triples and matched pairs for $A_3$-associative algebras. The paper defines the $A_3$-associative Yang-Baxter equation $AY(r)$ and shows that skew-symmetric solutions induce triangular $A_3$-associative bialgebras via coboundary maps $\Delta_r$, with relative Rota-Baxter operators providing operator realizations of these solutions. Collectively, these results generalize infinitesimal and operatorial approaches to bialgebras from associative and left-symmetric contexts to the broader $A_3$-associative framework, enriching the algebraic toolbox for Lie-admissible structures.

Abstract

$A_3$-associative algebra is a generalization of associative algebra and is one of the four remarkable types of Lie-admissible algebras, along with associative algebra, left-symmetric algebra and right-symmetric algebra. This paper develops bialgebra theory for $A_3$-associative algebras. We introduce Manin triples and bialgebras for $A_3$-associative algebras, prove their equivalence using matched pairs of $A_3$-associative algebras, and define the $A_3$-associative Yang-Baxter equation and triangular $A_3$-associative bialgebras. Additionally, we introduce relative Rota-Baxter operators to provide skew-symmetric solutions of the $A_3$-associative Yang-Baxter equation.

Theorems & Definitions (83)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Proposition 2.9
• Definition 2.10