Quantum Lines over Non-Commutative Cosemisimple Hopf Algebras

Let $R_q$ denote the N-dimensional quantum line $R_q=k[y]/(y^N)$ and let $H$ be an Hopf algebra. In this paper the following problems are considered: \itemitem{$\bullet$} classifying all possible biproducts $R_q\#H$; \itemitem{$\bullet$} deciding when $R_q\#H$ is selfdual; \itemitem{$\bullet$} classifying all Hopf algebras $A$ with coradical $H$ such that ${\rm gr}(A)$ is isomorphic to a given $R_q\#H$. These problems are solved in the case in which $H$ is one of the noncommutative cosemisimple Hopf algebras $\widehat{\scr D}_{2n}$ ($n\geq 3$), $\widehat{\scr T}_{4m}$ ($m\geq 2$), $\widehat{\scr A}_{4m}$ ($m\geq 3$), $\widehat{\scr B}_{4m}$ ($m\geq 2$), as introduced by the third author in [ New trends in Hopf algebra theory (La Falda, 1999), 195--214, Contemp. Math., 267, Amer. Math. Soc., Providence, RI, 2000]. As an application, it is proved that, over an algebraically closed field of characteristic different from $2$, there exist precisely two isomorphism classes of Hopf algebras of dimension $16$, whose coradical is a non-cocommutative Hopf subalgebra of dimension $8$. Together with the existing classification of pointed and semisimple Hopf algebras of dimension $16$, this completes the classification of $16$-dimensional Hopf algebras with the Chevalley property.