${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$的BLM实现
BLM realization for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$
摘要：
1990年，Beilinson-Lusztig-MacPherson（BLM）通过量子Schur代数的几何设置发现了量子$\frak{gl}_n$的实现\cite [5.7]{BLM}。我们将把他们的结果推广到经典仿射情形。更准确地说，我们首先使用Ringel-Hall代数构造${\mathcal U}(\hat{\frak{gl}}_n)$的积分形式${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$，其中${\mathcal U}(\hat{\frak{gl}}_n)$是循环代数$\hat{\frak{gl}}_n:=\frak{gl}_n(\mathbb Q)\otimes\mathbb Q[t,t^{-1}]$的泛包络代数。然后，我们建立了经典仿射Schur代数的乘法的稳定性。这种稳定性导致${\mathcal U}(\hat{\frak{gl}}_n)$和${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$的BLM实现。特别地，我们得出结论：${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$是${\mathcal U}(\hat{\frak{gl}}_n)$的$\mathbb Z$-Hopf子代数.作为奖励，这种方法导致一个明确的$\mathbb Z$ -基础${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$，它产生明确的乘法公式之间的发电机和基础元素的${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$。作为应用，我们证明了从${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$到$\mathbb Z$上仿射Schur代数的自然代数同态是满射的.
英文摘要：
In 1990, Beilinson-Lusztig-MacPherson (BLM) discovered a realization \cite[5.7]{BLM} for quantum $\frak{gl}_n$ via a geometric setting of quantum Schur 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 ${\mathcal U}(\hat{\frak{gl}}_n)$, where ${\mathcal U}(\hat{\frak{gl}}_n)$ is the universal enveloping algebra of the loop algebra $\hat{\frak{gl}}_n:=\frak{gl}_n(\mathbb Q)\otimes\mathbb Q[t,t^{-1}]$. We then establish the stabilization property of multiplication for the classical affine Schur algebras. This stabilization property leads to the BLM realization of ${\mathcal U}(\hat{\frak{gl}}_n)$ and ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$. In particular, we conclude that ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$ is a $\mathbb Z$-Hopf subalgebra of ${\mathcal U}(\hat{\frak{gl}}_n)$. As a bonus, this method leads to an explicit $\mathbb Z$-basis for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$, and it yields explicit multiplication formulas between generators and basis elements for ${\mathcal U}_{\mathbb Z}(\hat{\frak{gl}}_n)$. As an application, we will 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.
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