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

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.