From racks to pointed Hopf algebras
type: Item
instance of: scholarly article
title: From racks to pointed Hopf algebras
zbMATH Open document ID: 1032.16028
DOI: 10.1016/S0001-8708(02)00071-3
author: Matías Graña
author: Nicolás Andruskiewitsch
published in: Advances in Mathematics
publication date: 2003-09-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0202084
review text: 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.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 1985908
zbMATH Keywords: pointed Hopf algebras
zbMATH Keywords: finite simple racks
zbMATH Keywords: quandles
zbMATH Keywords: braided vector spaces
zbMATH Keywords: Nichols algebras
zbMATH Keywords: cohomology groups
zbMATH Keywords: cocycles
reviewed by: Sorin Dascalescu
MaRDI profile type: Publication
arXiv ID: math/0202084
cites work: Q5301202
cites work: Q4944224
cites work: Finite quantum groups and Cartan matrices
cites work: Q3153375
cites work: Q3203214
cites work: Twisted quandle homology theory and cocycle knot invariants
cites work: Extensions of Quandles and Cocycle Knot Invariants
cites work: State-sum invariants of knotted curves and surfaces from quandle cohomology
cites work: Quandle homology groups, their Betti numbers, and virtual knots
cites work: Q4459926
cites work: Braids and self-distributivity
cites work: On some unsolved problems in quantum group theory
cites work: On rack cohomology.
cites work: Indecomposable set-theoretical solutions to the quantum Yang-Baxter equation on a set with a prime number of elements
cites work: Set-theoretical solutions to the quantum Yang-Baxter equation
cites work: RACKS AND LINKS IN CODIMENSION TWO
cites work: Q4227181
cites work: A freeness theorem for Nichols algebras
cites work: A classifying invariant of knots, the knot quandle
cites work: Simple quandles
cites work: Q4000283
cites work: The Betti numbers of some finite racks
cites work: On the set-theoretical Yang-Baxter equation
cites work: On Hopf algebras with positive bases
cites work: Quasi-triangular structures on Hopf algebras with positive bases
cites work: Some calculations of cohomology groups of finite Alexander quandles
cites work: Indecomposable Coalgebras, Simple Comodules, and Pointed Hopf Algebras
cites work: Pointed indecomposable Hopf algebras over Coxeter groups.
cites work: Q4819420
cites work: Bialgebras of type one*
cites work: Homotopical algebra
cites work: Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation.
cites work: Survey of braided Hopf algebras
description: scientific article