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

Abstract

In 1990, Beilinson-Lusztig-MacPherson (BLM) discovered a realization \cite[5.7]{BLM} for quantum ${\mathfrak{gl}}_n$ 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 ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$ of $\mathcal{U}({\widehat{\mathfrak{gl}}}_n)$, where $\mathcal{U}({\widehat{\mathfrak{gl}}}_n)$ is the universal enveloping algebra of the loop algebra ${\widehat{\mathfrak{gl}}}_n:={\mathfrak{gl}}_n(\mathbb{Q})\otimes \mathbb{Q}[t,t^{-1}]$. We then establish the stabilization property of multiplication for the classical affine Schur algebras. This stabilization property leads to the BLM realization of $\mathcal{U}({\widehat{\mathfrak{gl}}}_n)$ and ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$. In particular, we conclude that ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$ is a $\mathbb{Z}$-Hopf subalgebra of $\mathcal{U}({\widehat{\mathfrak{gl}}}_n)$. As a bonus, this method leads to an explicit $\mathbb{Z}$-basis for ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$, and it yields explicit multiplication formulas between generators and basis elements for ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$. As an application, we will prove that the natural algebra homomorphism from ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$ to the affine Schur algebra over $\mathbb{Z}$ is surjective.

33 pages

Cited by in corpus (1)

• BLM realization for the integral form of quantum ${\mathfrak{gl}}_n$