BGP-Reflection Functors and Lusztig's Symmetries of Modified Quantized Enveloping Algebras

TL;DR

This paper connects BGP-reflection functors with Lusztig's symmetries in quantized enveloping algebras, providing a new realization of these symmetries via a modified Ringel-Hall algebra.

Contribution

It introduces a modified Ringel-Hall algebra and demonstrates how to realize Lusztig's symmetries on the modified quantum algebra using BGP-reflection functors.

Findings

Realization of Lusztig's symmetries through BGP-reflection functors.

Construction of a modified Ringel-Hall algebra $ ilde{ extbf{H}}$.

Verification of braid relations for the symmetries.

Abstract

Let $\mathbf{U}$ be the quantized enveloping algebra and $\dot{\mathbf{U}}$ its modified form. Lusztig gives some symmetries on $\mathbf{U}$ and $\dot{\mathbf{U}}$. Since the realization of $\mathbf{U}$ by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on $\mathbf{U}$ and their important properties, for instance, the braid relations. In this paper, we define a modified form $\dot{\mathcal{H}}$ of the Ringel-Hall algebra and realize the Lusztig's symmetries on $\dot{\mathbf{U}}$ by applying the BGP-reflection functors to $\dot{\mathcal{H}}$.