Integral affine Schur-Weyl reciprocity

TL;DR

This paper constructs integral forms of certain quantum affine algebras and proves their surjectivity onto affine Schur algebras, advancing the understanding of integral structures in quantum group theory.

Contribution

It introduces integral forms of the double Ringel--Hall algebra and the universal enveloping algebra of the loop algebra, establishing surjective homomorphisms to affine Schur algebras.

Findings

Constructed an integral form of the modified quantum affine algebra.

Proved the surjectivity of the algebra homomorphism to affine quantum Schur algebra.

Developed an integral form of the universal enveloping algebra of the loop algebra.

Abstract

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