Finite quantum groups and Cartan matrices
type: Item
instance of: scholarly article
title: Finite quantum groups and Cartan matrices
zbMATH Open document ID: 1007.16027
author: Nicolás Andruskiewitsch
author: Hans-Jürgen Schneider
published in: Advances in Mathematics
publication date: 2003-03-18T00:00:00Z
review text: This paper considers a class of Hopf algebras over an algebraically closed field \(k\) having as an invariant a generalized Cartan matrix. Let \(\Gamma\) be a fixed finite Abelian group, \(V\) a finite-dimensional Yetter-Drinfeld module over \(k\Gamma\) with a \(k\)-basis \(\{x_1,\dots,x_\theta\}\). The braiding \(c\colon V\otimes V\to V\otimes V\) is given with respect to the basis \(\{x_i\otimes x_j\}\) by \(c(x_i\otimes x_j)=b_{ij}x_j\otimes x_i\). If for all \(i,j\), \(b_{ii}\neq 1\) and there exists \(a_{ij}\in\mathbb{Z}\) such that \(b_{ij}b_{ji}=b^{a_{ij}}_{ii}\), then the braiding given by a matrix \({\mathbf b}=(b_{ij})_{1\leq i,j\leq\theta}\) whose entries are roots of unity is called to be of Cartan type, where \((a_{ij})\) is a generalized Cartan matrix. A Cartan type braiding \(\mathbf b\) is of FL-type if there exist positive integers \(d_1,\dots,d_\theta\) such that \(d_ia_{ij}=d_ja_{ji}\) and there exists \(q\in k\) such that \(b_{ij}=q^{d_ia_{ij}}\) for all \(i,j\). Furthermore, we say that a braiding \(\mathbf b\) is locally of FL-type if any principal \(2\times 2\) submatrix of \(\mathbf b\) gives a braiding of FL-type. For any Yetter-Drinfeld module \(V\) over \(\Gamma\), denote by \({\mathcal B}(V)\) the Nichols algebra whose space of primitive elements is isomorphic to the Yetter-Drinfeld module \(V\). Let \({\mathbf b}=(b_{ij})\) be a braiding of Cartan type, corresponding to \(c\colon V\otimes V\to V\otimes V\), and assume that \(b_{ij}\) has odd order for all \(i,j\). Let \(\mathcal X\) be the set of connected components of the Dynkin diagram corresponding to \(\mathbf b\). For each \(I\in{\mathcal X}\), let \({\mathfrak g}_I\) be the Kac-Moody Lie algebra corresponding to the generalized Cartan matrix \((a_{ij})_{i,j\in I}\) and \({\mathfrak n}_I\) be the Lie subalgebra of \({\mathfrak g}_I\) spanned by all its positive roots. If \(\mathbf b\) is of finite Cartan type, then \({\mathcal B}(V)\) is finite-dimensional, and if moreover 3 does n…
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1525150
zbMATH Keywords: Hopf algebras
zbMATH Keywords: generalized Cartan matrices
zbMATH Keywords: Yetter-Drinfeld modules
zbMATH Keywords: braidings
zbMATH Keywords: Nichols algebras
zbMATH Keywords: primitive elements
zbMATH Keywords: Dynkin diagrams
zbMATH Keywords: Kac-Moody Lie algebras
zbMATH Keywords: root systems
zbMATH Keywords: finite-dimensional coradically graded Hopf algebras
MaRDI profile type: Publication
full work available at URL: aima.1999.1880
OpenAlex ID: W2043346400
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DOI: 10.1006/AIMA.1999.1880
description: scientific article; zbMATH DE number 1525150