Rank 2 Nichols algebras with finite arithmetic root system (Q2478181)

Let \(\mathbf q = (q_{ij})_{1\leq i,j\leq 2}\) be a complex matrix such that \(q_{ij}\) and \(q_{ii}\neq 1\) are roots of 1, for all \(i,j\). Consider a vector space \(V\) with basis \(x_1, x_2\) and \(c\in GL(V\otimes V)\) given by \(c(x_i\otimes x_j) = q_{ij} x_j\otimes x_i\) for all \(i,j\); then \(c\) is a solution of the braid equation and we can consider the associated Nichols algebra \(\mathcal B(V)\). In this article, the author classifies all matrices \(\mathbf q\) such that the Nichols algebra \(\mathcal B(V)\) is finite-dimensional. For background and relevance of this result, we refer to the review of [\textit{I. Heckenberger}, Adv. Math. 220, No. 1, 59--124 (2009; Zbl 1176.17011)]. The case of rank 2 is crucial in for the general case, but also for the classification theorem in [N. Andruskiewitsch and H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math., to appear (2009)].