Schur–Weyl duality for infinitesimal q-Schur algebras sq(2,r)1

Using the result of [S.R. Doty, D.K. Nakano, K.M. Peters, Polynomial representations of Frobenius kernels of GL2, in: Contemp. Math., vol. 194, 1996, pp. 57–67; S. König, C. Xi, When is a cellular algebra quasi-hereditary? Math. Ann. 315 (1999) 281–293], we prove that a non-semisimple infinitesimal Schur algebra s(2,r)1 is not cellular. Furthermore, we determine the structure of the endomorphism ring of tensor space as a module for the infinitesimal Schur algebra s(2,r)1, up to Morita equivalence. Both results generalize to the quantum case.