A Cacti theoretical interpretation of the axioms of bialgebras and H-module algebras

TL;DR

This paper provides a theoretical framework linking Cacti algebra axioms with bialgebras and H-module algebras, translating complex algebraic structures into a unified interpretative language.

Contribution

It establishes a dictionary connecting Cacti algebra axioms with algebraic structures on associative algebras and bialgebras, enabling new insights and applications.

Findings

Established a correspondence between Cacti algebra axioms and algebraic structures.

Translated Cacti algebra maps into Hochschild cohomology contexts.

Provided examples demonstrating applications of the theoretical framework.

Abstract

We establish a dictionary between the Cacti algebra axioms on a Cacti algebra structure with underlying free associative algebra, under suitable good behavior with degrees. Using these ideas, for an associative algebra $A$ and a bialgebra $H$, we also translate Cacti algebra maps $\Omega(H)\to C^\bullet(A)$ (where $\Omega(H)$ stands for the cobar construction on $H$ and $C^\bullet(A)$ is the Hochschild cohomology complex) with $H$-module algebra structures on $A$, and illustrate with examples of applications.