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

Let U be a quantized enveloping algebra and  its modified form.Lusztig gives some symmetries on U and.In view of the realization of 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 U and their important properties,for instance,the braid relations.In this paper,we define a modified form ■ of the Ringel-Hall algebra and realize the Lusztig's symmetries on  by applying the BGP-reflection functors to ■.

Let U be a quantized enveloping algebra and U its modified form. Lusztlg gives some symmetries on U and U. In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall Mgebra, one can apply the BGP--efiection functors to the double Ringel-HM1 algebra to obtain Lusztig＇s symmetries on U and their important properties, for instance, the braid relations. In this paper .we define a modified form of the Ringe Hall. algebra and realize the Lusztig＇s symmetries on U by applying the BGP-reflection functors to H.