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 Ω(H) → C•(A) (where Ω(H) stands for the cobar construction on H and C•(A) is the Hochschild cohomology complex) with H-module algebra structures on A, and illustrate with examples of applications.