2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/105122The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding results for them.Final version, to appear in Adv. MathQuantum AlgebraRepresentation Theorytext