The Lusztig Symmetries And Algebra Automorphisms Of U_q(osp(1|2r))

Let(?) be a finite dimensional semisimple Lie algebra with Cartan matrix (a_ij)_n×n,then we have Drinfeld-Jimbo quantized eveloping algebra U_q((?)).To consider the PBW basis of U_q((?)),then the canonical basis,Lusztig introduced a series of automorphisms of U_q((?)),which are called Lusztig symmetries.The fundamental result is that they satisfy the braid group relations.Lusztig symmetries are so important for quantized enveloping algebra that many people have investigated them.However,up to now,nobody has constructed the Lusztig symmetries of quantized enveloping superalgebra.In this thesis,we first construct algebra automophisms T_i(1≤i≤r) of Z₂-graded Hopf algebra U_q(osp(1|2r)).It is shown that they satisfy braid group relations.In this thesis,these algebra automorphisms are also called Lusztig symmetries,secondly,We construct PBW basis of U_q(osp(1|2r)) by Lusztig symmetries. Finally,we investegate the algebra automorphisms of U_q(osp(1|2r)) by Lusztig symmetries and PBW basis.