Irreducible modules of finite dimensional quantum algebras of type A at roots of unity

Properly specializing the parameters contained in the maximal cyclic representation of the nonrestricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. In this case, the representation is no longer irreducible. We show that the submodule generated by the primitive vector is the unique irreducible submodule and can be identified with an irreducible highest weight module of the finite dimensional A-type quantum algebra, which is defined as the subalgebra of the restricted quantum algebra at roots of unity.