Integral affine Schur–Weyl reciprocity

Abstract

Let D▵(n) be the double Ringel–Hall algebra of the cyclic quiver △(n) and let D▵̇(n) be the modified quantum affine algebra of D▵(n). We will construct an integral form D▵̇(n)Z for D▵̇(n) such that the natural algebra homomorphism from D▵̇(n)Z to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral form UZ(gl̂n) of the universal enveloping algebra U(gl̂n) of the loop algebra gl̂n=gln(Q)⊗Q[t,t−1], and prove that the natural algebra homomorphism from UZ(gl̂n) to the affine Schur algebra over Z is surjective. In a subsequent paper (Fu [10]), we will use affine Schur algebras to give BLM realization of UZ(gl̂n), and this enables us to give a new proof of the statements about UZ(gl̂n) given in this paper.