Abstract
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-Drinfel’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.
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This work was supported by Henri Lebesgue Centre (University of Nantes), and by the program ANR-11-LABX-0020-01.
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Lebed, V., Wagemann, F. Representations of Crossed Modules and Other Generalized Yetter-Drinfel’d Modules. Appl Categor Struct 25, 455–488 (2017). https://doi.org/10.1007/s10485-015-9421-z
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DOI: https://doi.org/10.1007/s10485-015-9421-z
Keywords
- Yang-Baxter equation
- Yetter-Drinfel’d module
- Self-distributivity
- Crossed module of groups
- Leibniz algebra
- Monoidal category