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

Victoria Lebed (1)

1 OCAMI - Osaka City university Advanced Mathematical Institute [Osaka]

Bialgebras and Hopf (bi)modules are examples of algebraic structures involving several interacting operations. The multi-operation setting significantly increases the complexity of these structures and their homology. In the present paper we develop the machinery of braided systems, tailored for handling multi-operation structures. Our construction is general enough to include as particular cases the examples above (as well as, for instance, Poisson algebras and Yetter-Drinfel'd modules, treated in separate publications). At the same time, graphical tools allow a concrete and efficient exploration of braided systems. Gerstenhaber-Schack, Panaite-Stefan and Ospel-Taillefer (co)homology theories for bialgebras and Hopf (bi)modules, as well as the Heisenberg double, and the algebras X, Y and Z of Cibils-Rosso and Panaite, naturally appear in our braided setting. This new interpretation offers a conceptual explication, a generalization and a simplified proof of several related algebraic phenomena.

Mots clés: braided systems; braided homology; Hopf algebras; Hopf (bi)modules; Heisenberg double; (generalized) crossed products; bialgebra homology; distributive laws; multi-quantum shuffle algebra

Domaines: Algèbres quantiques [math.QA]; K-théorie et homologie [math.KT]; Catégories et ensembles [math.CT]

Victoria Lebed. Braided Systems: a Unified Treatment of Algebraic Structures with Several Operations. 2013. ⟨hal-00820327v2⟩