Braided Systems, Multi-Braided Tensor Products and Bialgebra Homologies

Victoria Lebed (1)

1 IMJ - Institut de Mathématiques de Jussieu

We introduce the notion of braided system and develop its representation and homology theories. The case of braided systems of associative algebras is presented in detail, giving an efficient tool for studying multi-braided tensor products of algebras and their actions. Braided systems encoding the structures of generalized crossed product and bialgebra are considered. For the latter, Hopf modules are identified as the corresponding multi-braided modules, Heisenberg double as the corresponding multi-braided tensor product of algebras, and Gerstenhaber-Schack and Panaite-Stefan (co)homologies as particular cases of multi-braided (co)homologies. This ''braided'' interpretation offers a conceptual explication and a simplified proof of several algebraic phenomena concerning the structures above.

Mots clés: braided systems; braided homology; multi-braided modules; Hopf algebras; bialgebra homology; Hopf modules; Heisenberg double; module algebras; (generalized) crossed products; multi-braided algebra tensor products; 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, Multi-Braided Tensor Products and Bialgebra Homologies. 2013. ⟨hal-00820327v1⟩