47 research outputs found
Braided Hopf algebras arising from matched pairs of groups
Let k be a field. Let also (F, G) be a matched pair of groups. We give
necessary and sufficient conditions on a pair (\sigma, \tau) of 2-cocycles in
order that the crossed product algebra and the crossed coproduct coalgebra
k^G{}^{\tau}#_{\sigma} kF combine into a braided Hopf algebra. We also discuss
diagonal realizations of such braided Hopf algebras in the category of
Yetter-Drinfeld modules over a finite group.Comment: 25 pages, to appear in J. of Pure and Applied Algebra, appendix added
at the end of the paper by suggestion of the refere
From racks to pointed Hopf algebras
A fundamental step in the classification of finite-dimensional complex
pointed Hopf algebras is the determination of all finite-dimensional Nichols
algebras of braided vector spaces arising from groups. The most important class
of braided vector spaces arising from groups is the class of braided vector
spaces (CX, c^q), where C is the field of complex numbers, X is a rack and q is
a 2-cocycle on X with values in C^*. Racks and cohomology of racks appeared
also in the work of topologists. This leads us to the study of the structure of
racks, their cohomology groups and the corresponding Nichols algebras. We will
show advances in these three directions. We classify simple racks in
group-theoretical terms; we describe projections of racks in terms of general
cocycles; we introduce a general cohomology theory of racks contaninig properly
the existing ones. We introduce a "Fourier transform" on racks of certain type;
finally, we compute some new examples of finite-dimensional Nichols algebras.Comment: 54 pages. Several minor corrections. Some references added. Same
version as will appear in Adv. Mat