Schur algebras are certain finite-dimensional algebras that completely
control the
polynomial representation theory of the general linear groups over an infinite
field.
Infinitesimal Schur algebras are truncated versions of the classical Schur
algebras
which control the polynomial representation theory of the Frobenius kernels
of
general linear groups. In this paper we use some elementary results on
symmetric powers
to classify the semisimple Schur algebras. We then classify the semisimple
infinitesimal Schur algebras as well.