‎In this paper‎, ‎we prove that the category of crossed polymodules (i.e‎. ‎crossed modules of polygroups) and their morphisms is finitely complete‎. ‎We‎, ‎therefore‎, ‎generalize the group theoretical case of this completeness property of crossed modules‎.

We introduce a representation theory of q-Lie algebras defined earlier in [DG1], [DG2], formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules.

In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the...

in this paper, we present algebroids and crossed modules of algebroids. we also define pullbackcrossed module of algebroids.

1 Crossed Modules and Cat1-Groups 4 1.1 Pre-crossed and Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Examples of Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Properties of Crossed Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Sub-crossed modules . . . . . . . . . . . . . . . . . . . . . ....

In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel′d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibi...