Tensor products and filtrations of principal tilting modules for quantum groups. (Q2732552)

Let \(A=\mathbb{Z}[v]_\Omega\) be the localization of \(\mathbb{Z}[v]\) at its ideal \(\Omega\) generated by \(v-1\) and an odd prime \(p\), and \(U\) the quantum algebra over \(A\). Let \(\phi_p\) be the \(p\)-th cyclotomic polynomial, \(B=A/(\phi_p)\), and \(\Gamma\) the completion at the ideal \((\xi-1)\), where \(\xi\) is a \(p\)-th primitive root. For \(\lambda\in X^+\), \(M_\Gamma(\lambda)\) denotes the indecomposable tilting \(U_\Gamma\)-module with highest weight \(\lambda\). In this paper the author proves a tensor product theorem for indecomposable tilting \(U_\Gamma\)-modules, and studies the filtration of indecomposable tilting \(U_\Gamma\)-modules when \(p\geq 2h-2\). As an example, he gives the decomposition pattern of good filtrations of indecomposable tilting \(U_\Gamma\)-modules for rank \(1\) quantum groups and \(A_2\) type quantum groups.