BLM realization for the integral form of quantum ${\mathfrak{gl}}_n$

Abstract

Let $\mathbf{U}(n)$ be the quantum enveloping algebra of ${\mathfrak{gl}}_n$ over $\mathbb{Q}(v)$, where $v$ is an indeterminate. We will use $q$-Schur algebras to realize the integral form of $\mathbf{U}(n)$. Furthermore we will use this result to realize quantum ${\mathfrak{gl}}_n$ over $k$, where $k$ is a field containing an l-th primitive root $\epsilon$ of 1 with $l\ge 1$ odd.