Quantum lines over non-cocommutative cosemisimple Hopf algebras.
Let \(q\) be a primitive root of 1 of degree \(n\) in a field \(k\) and \(R_q\) the graded \(k\)-algebra \(k[Y]/(y^n)\) endowed with a coalgebra structure such that \(\tfrac{y^j}{(j)_q}\), \(1\leqslant j\leqslant n-1\), forms a divided power sequence. Let \(H\) be one of the following non-commutative cosemisimple Hopf algebras \[ \widehat{\mathcal D}_{2n},\;n\geqslant 3,\quad\widehat{\mathcal T}_{4m},\;m\geqslant 2,\quad\widehat{\mathcal A}_{4m},\;m\geqslant 3,\quad\widehat{\mathcal B}_{4m},\;m\geqslant 2 \] from the paper of \textit{A. Masuoka} [Isr. J. Math. 92, No. 1-3, 361-373 (1995; Zbl 0839.16036)]. There is given a classification of biproducts \(R_q\#H\) as graded and ungraded Hopf algebras. In particular there is given a classification of Hopf algebras \(A\) with coradical \(H\) such that \(\text{gr\,}A\) is isomorphic to \(R_q\#H\). The problem of a selfduality of \(R_q\#H\) is also solved.
scientific article; zbMATH DE number 2055694
scientific article