Integral affine Schur-Weyl reciprocity

Abstract

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

20 pages