From racks to pointed Hopf algebras
The paper proposes an approach to the classification of finite dimensional pointed Hopf algebras over the field \(\mathbb{C}\) of complex numbers starting from racks. Racks have been considered in the literature mainly as a way to produce knot invariants. A rack is a set \(X\) with an operation \(\triangleright\colon X\times X\to X\) such that for any \(i\in X\) the map \(j\mapsto i\triangleright j\) is a bijection of \(X\), and \(i\triangleright(j\triangleright k)=(i\triangleright j)\triangleright(i\triangleright k)\) for any \(i,j,k\in X\). Epimorphisms of racks are described by general cocycles. Modules over a rack are defined and studied. An explicit classification of all finite simple racks is given in group theoretical terms. A general cohomology theory for racks is proposed. A braided vector space is associated to a finite rack and a 2-cocycle with values in \(\mathbb{C}^*\). A main problem discussed in the paper is to decide whether the Nichols algebra of such a braided vector space is finite dimensional. Some reductions to Nichols algebras of diagonal type are done by using some kind of Fourier transform. Several new examples of finite dimensional Nichols algebras are obtained.
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