Homology, Homotopy and Applications

Volume 19 (2017)

Number 2

Braided systems: a unified treatment of algebraic structures with several operations

Pages: 141 – 174

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a9


Victoria Lebed (Hamilton Mathematics Institute, Trinity College, Dublin, Ireland)


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–Ştefan, 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.


braided system, braided homology, Hopf algebra, Hopf (bi)module, Heisenberg double, crossed product, bialgebra homology, distributive law, multi-quantum shuffle algebra

2010 Mathematics Subject Classification

16E40, 16T05, 16T10, 16T25, 18D10

Received 24 April 2016

Received revised 18 January 2017

Published 18 October 2017