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

TL;DR

This paper proves the equivalence of two integral forms of the universal enveloping algebra of the affine Lie algebra ${\widehat{\frak{gl}}_n}$ and constructs a subalgebra related to hyperalgebras using Ringel--Hall algebras and affine Schur algebras.

Contribution

It establishes the coincidence of Garland's and Ringel--Hall algebra-based integral forms and constructs a new subalgebra of the affine quantum group related to hyperalgebras.

Findings

Proves Garland integral form coincides with Ringel--Hall algebra construction.

Constructs a subalgebra $\mathtt{u}_\vartriangle(n)_h$ of the affine algebra.

Provides a realization of $\mathtt{u}_\vartriangle(n)_h$ via affine Schur algebras.

Abstract

Let $\widetilde{\mathcal U}_{\mathbb Z}({\widehat{\frak{gl}}_n})$ be the Garland integral form of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ introduced by Garland \cite{Ga}, where ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ is the universal enveloping algebra of ${\widehat{{\frak{gl}}}_n}$. Using Ringel--Hall algebras, a certain integral form ${\mathcal U}_{\mathbb Z}(\widehat{{\frak{gl}}}_n)$ of ${\mathcal U}(\widehat{{\frak{gl}}}_n)$ was constructed in \cite{Fu13}. We prove that the Garland integral form $\widetilde{\mathcal U}_{\mathbb Z}({\widehat{{\frak{gl}}}_n})$ coincides with ${\mathcal U}_{\mathbb Z}(\widehat{{\frak{gl}}}_n)$. Let ${\mathpzc k}$ be a commutative ring with unity and let ${\mathcal U}_{\mathpzc k}(\widehat{{\frak{gl}}}_n)={\mathcal U}_{\mathbb Z}(\widehat{{\frak{gl}}}_n)\otimes{\mathpzc k}$. For $h\geq 1$, we use Ringel--Hall algebras to construct a certain subalgebra, denoted by ${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$, of ${\mathcal U}_{\mathpzc k}(\widehat{{\frak{gl}}}_n)$. The algebra ${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$ is the affine analogue of ${\mathtt{u}}({{\frak{gl}}}_n)_h$, where ${\mathtt{u}}({{\frak{gl}}}_n)_h$ is a certain subalgebra of the hyperalgebra associated with ${\frak{gl}}_n$ introduced by Humhpreys \cite{Hum}. The algebra ${\mathtt{u}}({{\frak{gl}}}_n)_h$ plays an important role in the modular representation theory of ${\frak{gl}}_n$. In this paper we give a realization of ${{\mathtt{u}}}_{\!\vartriangle\!}(n)_h$ using affine Schur algebras.