The Green ring of a family of copointed Hopf algebras

The copointed liftings of the Fomin-Kirillov algebra ${\mathcal{F}\mathcal{K}}_3$ over the algebra of functions on the symmetric group ${\mathbb{S}}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a generic parameter are Morita equivalent to the non-simple blocks of well-known Hopf algebras: the Drinfeld doubles of the Taft algebras and the small quantum groups $u_q({\mathfrak{s}\mathfrak{l}}_2)$. The indecomposable modules over these were classified independently by Chen, Chari–Premet and Suter. Consequently, we obtain the indocomposable modules over the generic liftings of ${\mathcal{F}\mathcal{K}}_3$. We decompose the tensor products between them into the direct sum of indecomposable modules. We then deduce a presentation by generators and relations of the Green ring.