On the hyperalgebra of the loop algebra $\hat{gl}_n$

Let $\tilde{U}_{Z}(\hat{gl}_n)$ be the Garland integral form of $U(\hat{gl}_n)$ introduced by Garland, where $U(\hat{gl}_n)$ is the universal enveloping algebra of $\hat{gl}_n$. Using Ringel-Hall algebras, one can naturally construct an integral form, denoted by $U_Z(\hat{gl}_n)$, of $U(\hat{gl}_n)$.

We prove that $\tilde{U}_{Z}(\hat{gl}_n)$ coincides with $U_Z(\hat{gl}_n)$.

Let $\kappa$ be a commutative ring with unity.

Assume $p=\text{char}\kappa>0$.

We call $U_{\kappa}(\hat{gl}_n):=U_Z(\hat{gl}_n)\otimes\kappa$ the hyperalgebra of $\hat{gl}_n$.

For $h\geq 1$, we use Ringel--Hall algebras to construct a certain subalgebra, denoted by $U_{\Delta}(n)_h$, of $U_{\kappa}(\hat{gl}_n)$. The algebra $U_{\Delta}(n)_h$

is the affine analogue of the restricted enveloping

algebra of $gl_n$ over ${\mathbb F}_p$.

We will give a realization of $U_{\Delta}(n)_h$ for each $h\geq 1$. Using $U_{\Delta}(n)_h$, we construct a certain subalgebra, denoted by $U_{\Delta}(n,r)_h$, of affine Schur algebras over $\kappa$. The algebra $U_{\Delta}(n,r)_h$ is the affine analogue of little Schur algebras.