The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures supergravity rely on a pairing – embedding between Leibniz and algebra. Two such algebras, together with tensor, form triple called Lie-Leibniz triple, which modules are partic...

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. That is, the category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.