To a root system R and a choice of coefficients in a field K we associate a category 𝒳 of graded spaces with operators. For an arbitrary choice of coefficients we show that we obtain a semisimple category in which the simple objects are parametrized by their highest weight. Then we assume that the coefficients are given by quantum binomials associated to (K,q), where q is an invertible element in K. In the case that R is simply laced and (K,q) has positive (quantum) characteristic, we construct a Frobenius pullback functor and prove a version of Steinberg’s tensor product theorem for 𝒳. Then we prove that one can view the objects in 𝒳 as the semisimple representations of Lusztig’s quantum group associated to (R,K,q) (for q=1 we obtain semisimple representations of the hyperalgebra associated to (R,K)). Hence we obtain new proofs of the Frobenius and Steinberg theorems both in the representation theory of reductive algebraic groups and of quantum groups.