BGP-Reflection Functors and Lusztigs Symmetries of Modified Quantized Enveloping Algebras

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 Lusztigs 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 Lusztigs symmetries on $dot{mathbf{U}}$ by applying the BGP-reflection functors to $dot{mathcal{H}}$.