Solutions of associative Yang-Baxter equation and $D-$equation in low dimensions and associated Frobenius algebras and Connes cocycles

TL;DR

This paper explicitly solves associative Yang-Baxter and D-equations in low-dimensional associative algebras, constructing Frobenius algebras and Connes cocycles, and exploring related dendriform structures.

Contribution

It provides explicit solutions and constructions for associative Yang-Baxter and D-equations in low dimensions, linking them to Frobenius and Connes cocycle structures.

Findings

Explicit solutions for associative Yang-Baxter equations in 1 and 2 dimensions

Construction of Frobenius algebras via skew-symmetric solutions

Double constructions of Connes cocycles from symmetric solutions

Abstract

This work addresses some relevant characteristics of associative algebras in low dimensions. Especially, given 1 and 2 dimensional associative algebras, we explicitly solve associative Yang-Baxter equations and use skew-symmetric solutions to perform double constructions of Frobenius algebras. Besides, we determine related compatible dendriform algebras and solutions of their $ D- $equations. Finally, using symmetric solutions of the latter equations, we proceed to double constructions of corresponding Connes cocycles.