Braided Systems: a Unified Treatment of Algebraic Structures with Several Operations

TL;DR

This paper introduces braided systems as a unified framework to study algebraic structures with multiple interacting operations, simplifying and generalizing existing homology theories and representation tools.

Contribution

It develops a graphical machinery for braided systems that unifies and extends homology and representation theories of complex algebraic structures.

Findings

Unified approach to bialgebras and Hopf modules

Graphical tools facilitate analysis of algebraic structures

Generalizes standard homology theories and representation algebras

Abstract

Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling such multi-operation situations. Our construction covers the above examples (as well as Poisson algebras, Yetter--Drinfel$'$d modules, and several other structures, treated in separate publications). In spite of this generality, graphical tools allow an efficient study of braided systems, in particular of their representation and homology theories. These latter naturally recover, generalize, and unify standard homology theories for bialgebras and Hopf (bi)modules (due to Gerstenhaber--Schack, Panaite--{\c{S}}tefan, Ospel, Taillefer); and the algebras encoding their representation theories (Heisenberg double, algebras~$\mathscr X$, $\mathscr Y$, $\mathscr Z$ of Cibils--Rosso and Panaite). Our approach yields simplified and conceptual proofs of the properties of these objects.