We prove the existence of a basis of Poincaré-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V ⊂ P (R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple Uq(sl2)-modules.

We study noncommutative differential structures on the group of permutations S N , defined by conjugacy classes. The 2-cycles class defines an exterior algebra Λ N which is a super analogue of the Fomin-Kirillov algebra E N for Schubert calculus on the cohomology of the GL N flag variety. Noncom-mutative de Rahm cohomology and moduli of flat connections are computed for N < 6. We find that flat...

The Hochschild and (cotriple) cyclic homologies of crossed modules of (notnecessarily-unital) associative algebras are investigated. Wodzicki’s excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cycl...

For n ≥ 4, we describe Artin’s braid group on n strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations. Subject classification: Primary: 20F36 57M05 57M20 57M25; Secondary: 18D50 18G55 20C08 55P48

In this paper following Conduché’s, [12], way of defining 2-crossed modules which are equivalent to Carrasco-Cegarra’s, [10], 2-hyper crossed complexes, we define the three crossed module of groups as a model for 4-types.

The well-known notion of crossed module of groups is raised in this paper to the categorical level supported by the theory of categorical groups. We construct the cokernel of a categorical crossed module and we establish the universal property of this categorical group. We also prove a suitable 2-dimensional version of the kernelcokernel lemma for a diagram of categorical crossed modules. We th...

Let (H, σ) be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define Hσ , a sub-Hopf algebra of H, the finite dual of H. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we...

After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We p...