The classification of all crossed products \(H_4\#k[C_n]\). (Q2447908)

Let \(k\) be a field, \(A\) and \(H\) Hopf \(k\)-algebras, \(f\colon H\otimes_kH\to A\) a \(k\)-linear map and \(\diamond\colon H\otimes_kA\to A\) a weak action of \(H\) on \(A\). Then the \(k\)-vector space \(A\otimes_kH\) is the crossed product \(A\#_f^\diamond H\) of \(A\) with \(H\) as defined by \textit{N. Andruskiewitsch} and \textit{S. Natale}, [J. Math. Sci., Tokyo 6, No. 1, 181-215 (1999; Zbl 0934.16033)], and \((A,H,\diamond,f)\) is called a crossed system of Hopf algebras. Then the classification of all coalgebra split extensions of \(A\) by \(H\) reduces to the classification of all crossed products \(A\#_f^ \diamond H\). Let \(CS(A,H)\) be the set of all \((\diamond,f)\) such that \((A,H,\diamond,f)\) is a crossed system of Hopf algebras. A cohomologous equivalence relation \(\approx\) on \(CS(A,H)\) is defined so that \(H^2(H,A)=CS(A,H)/\approx\). Let \(C_{RP}(H,A)\) be the set of types of Hopf algebra isomorphisms of all crossed products \(A\#_f^\diamond H\) associated to all crossed systems \((A,H,\diamond,f)\), \(H_4\) the Sweedler \(4\)-dimensional Hopf algebra and \(C_n\) the cyclic group of order \(n\). Then using the computational approach introduced by \textit{A. L. Agore} et al. [in J. Algebra Appl. 12, No. 5, Paper No. 1250227 (2013; Zbl 1271.16035)], the authors classify all crossed products of Hopf algebras \(H_4\#k[C_n]\) by explicitly computing \(H^2(k[C_n],H_4)\) and \(C_{RP}(k[C_n],H_4)\). They are \(4n\)-dimensional quantum groups \(H_{4n,\lambda, t}\), parameterized by the set of all pairs \((\lambda,t)\) consisting of an arbitrary unitary map \(t\colon C_n\to C_2\) and an \(n\)-th root \(\lambda\) of \(1\) and \(-1\).