Stable Green ring of the Drinfeld doubles of the generalised Taft algebras (corrections and new results) (Q2272703)

Let \(d\geq 2\) and \(n\) be positive integers such that \(d\mid n\). Let \(k\) be an algebraically closed field whose characteristic does not divide \(d\), and let \(q\) be a primitive \(d\)-th root of unity in \(k\). The authors consider the Hopf algebra \(\Lambda_{n,d}\), whose co-opposite dual is the generalized Taft algebra generated by a grouplike element \(G\) of order \(n\), and a \((1,G)\)-skew-primitive element \(X\), such that \(X^d=0\) and \(GX=q^{-1}XG\). The aim of the paper is to investigate decompositions of tensor products of indecomposable modules over the Drinfeld double \(\mathcal{D}(\Lambda_{n,d})\) as direct sums of indecomposable modules. Some previous results of the authors [J. Pure Appl. Algebra 204, 413--454 (2006; Zbl 1090.16015)] are corrected and completed by using a new homological approach, based on exploiting stable module homomorphisms. A complete description of the stable Green ring of \(\mathcal{D}(\Lambda_{n,d})\) is given. The authors also classify endotrivial and algebraic \(\mathcal{D}(\Lambda_{n,d})\)-modules.