Integral affine Schur-Weyl reciprocity

摘要: Let ${\boldsymbol{\mathfrak D}_{\vartriangle}}(n)$ be the double Ringel--Hall algebra of cyclic quiver $\triangle(n)$ and let $\dot{\boldsymbol{\mathfrak modified quantum affine D}_{\vartriangle}}(n)$. We will construct an integral form $\dot{{\mathfrak for such that natural homomorphism from to Schur is surjective. Furthermore, we use Hall algebras ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ universal enveloping U}(\hat{\frak{gl}}_n)$ loop $\hat{\frak{gl}}_n=\frak{gl}_n({\mathbb Q})\otimes\mathbb Q[t,t^{-1}]$, prove U}_\mathbb Z(\hat{\frak{gl}}_n)$ over $\mathbb Z$