Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories

RESUMO

Let \(G\) be a finite non-abelian group, let \(\Phi\in Z^3(G,\mathbb C^\times)\) be normalized, and let \(V,W\) be finite-dimensional simple objects of \({}_G^G\mathcal{YD}^{\Phi}\). We classify, up to interchange, the braided-indecomposable pairs \((V,W)\) whose supports generate \(G\) and for which \(\mathcal B(V\oplus W)\) is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type \(A_2\), \(B_2\), or \(G_2\), and the dimension is determined explicitly. A new phenomenon occurs for \(\Gamma _2\): the twisted setting admits a family of type \(G_2\) absent from the ordinary \(\Gamma _2\) classification and we construct an explicit example over a non-abelian group of order \(16\).