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In 1990, Beilinson-Lusztig-MacPherson (BLM) discovered a realization\cite[5.7]{BLM} for quantum $\frak{gl}_n$ via a geometric setting of quantumSchur algebras. We will generailze their result to the classical affine case.More precisely, we first use Ringel-Hall algebras to construct an integral form${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ of ${\mathcalU}(\hat{\frak{gl}}_n)$, where ${\mathcal U}(\hat{\frak{gl}}_n)$ is theuniversal enveloping algebra of the loop algebra$\hat{\frak{gl}}_n:=\frak{gl}_n(\mathbb Q)\otimes\mathbb Q[t,t^{-1}]$. We thenestablish the stabilization property of multiplication for the classical affineSchur algebras. This stabilization property leads to the BLM realization of${\mathcal U}(\hat{\frak{gl}}_n)$ and ${\mathcal U}_{\mathbbZ}(\hat{\frak{gl}}_n)$. In particular, we conclude that ${\mathcal U}_{\mathbbZ}(\hat{\frak{gl}}_n)$ is a $\mathbb Z$-Hopf subalgebra of ${\mathcalU}(\hat{\frak{gl}}_n)$. As a bonus, this method leads to an explicit $\mathbbZ$-basis for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$, and it yieldsexplicit multiplication formulas between generators and basis elements for${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$. As an application, we will provethat the natural algebra homomorphism from ${\mathcal U}_{\mathbbZ}(\hat{\frak{gl}}_n)$ to the affine Schur algebra over $\mathbb Z$ issurjective.

BLM realization for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$