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

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.