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

Abstract

Let ${\mathbf U}(n)$ be the quantum enveloping algebra of ${\frak {gl}}_n$ 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 $\frak{gl}_n$ over $k$, where $k$ is a field containing an l-th primitive root $\varepsilon$ of 1 with $l\geq 1$ odd.