Finite quantum groups and Cartan matrices
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 not divide the order of \(b_{ii}\) for all \(i\) in a connected component of the Dynkin diagram of type \(G_2\), then \(\dim{\mathcal B}(V)=\prod_{I\in{\mathcal X}}N^{\dim{\mathfrak n}_I}_I\), where \(N_I=\text{ord}(b_{ij})\) for all \(i\in I\) and \(I\in{\mathcal X}\). Assume that \(\mathbf b\) is locally of FL-type and that, for all \(i\), the order of \(b_{ii}\) is relatively prime to 3 whenever \(a_{ij}=-3\) for some \(j\), and is different from 3, 5, 7, 11, 13, 17. If \({\mathcal B}(V)\) is finite-dimensional, then \(\mathbf b\) is of finite type. Let \(p\) be an odd prime number, \(\Gamma\) a finite direct sum of copies of \(\mathbb{Z}/(p)\) and \(V\) a finite-dimensional Yetter-Drinfeld module over \(\Gamma\) with braiding \(\mathbf b\). Assume that \(b_{ii}\neq 1\) for all \(i\). Then \(\mathbf b\) is of Cartan type. If \(\mathbf b\) is of finite type, then \({\mathcal B}(V)\) is finite-dimensional, and \(\dim{\mathcal B}(V)=p^M\), where \(M=\sum_{I\in{\mathcal X}}\dim{\mathfrak n}_I\) is the number of positive roots of the root system of \(\mathbf b\). If \({\mathcal B}(V)\) is finite-dimensional and \(p>17\), then \(\mathbf b\) is of finite type. For the case that \(\Gamma\simeq\mathbb{Z}/(p)\), where \(p\) is an odd prime number, this paper classifies the Nichols algebras of finite dimension over \(\Gamma\) and finite-dimensional coradically graded Hopf algebras with coradical isomorphic to \(k\Gamma\).
finite-dimensional coradically graded Hopf algebras
scientific article; zbMATH DE number 1525150
scientific article