From racks to pointed Hopf algebras

Details From racks to pointed Hopf algebras From racks to pointed Hopf algebras

2002-02-11

arxiv.org/abs/math/0202084v3

Adv. in Math. 178 (2), 177--243 (2003)

Mathematics

Quantum Algebra

54 pages. Several minor corrections. Some references added. Same version as will appear in Adv. Math

Scientific paper

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.

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