REPRESENTATIONS OF CROSSED MODULES AND OTHER GENERALIZED YETTER-DRINFEL'D MODULES

Victoria Lebed (1) , Friedrich Wagemann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfe' d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel'd (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators.

Mots clés: Yang-Baxter equation; Yetter-Drinfel'd module; Hopf algebra; self-distributivity; crossed module of groups; crossed module of racks; crossed module of Lie algebras; braided system; monoidal category; associator

Domaines: Algèbres quantiques [math.QA]; Catégories et ensembles [math.CT]; Topologie algébrique [math.AT]

Victoria Lebed, Friedrich Wagemann. REPRESENTATIONS OF CROSSED MODULES AND OTHER GENERALIZED YETTER-DRINFEL'D MODULES. Applied Categorical Structures, 2016, ⟨10.1007/s10485-015-9421-z⟩. ⟨hal-01197249⟩