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

Abstract

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

arXiv comment30 Pages

References in corpus (2)

• The Integral quantum loop algebra of ${\mathfrak{gl}}_n$
• BLM realization for ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$