Induced Modules And The First Cohomology Of Modular Lie Superalgebra (?)(2)

Lie superalgebra is an important branch of algebra,which is closely related to the theory of Lie algebra and is used to solve the problem of supersymmetry in physics.Therefore,Lie superalgebra is widely concerned by mathematicians and physicists.In this paper,we study the irreducibility of （?）（2）-modules of Lie superalgebras with chap F=p>2.First analyzed the（?）（2）zero parts of irreducible exactly the irreducibility of induction,prove that the irreducible（?）（2）-modules are irreducible （?）（2）₀-induced modudles of homomorphism image.Secondly,it is proved that all sim-ple modules of（?）（2）are isomorphic to the irreducible quotient module of its induced module,and the number and dimension of isomorphic classes of（?）（2）-irreducible mod-ules are determined.The Kac-Weisfeiler conjecture of（?）（2）is obtained.Finally,by calculating the weight derivates of（?）（2）induced modules,the first cohomology is obtained.The thesis consists of four chapters:Chapter 1 introduces the research background,significance,present situation and main conclusions of this paper.Chapter 2 introduces the basic concepts of modular Lie algebras,modular Lie superalgebras and restricted modular Lie superalgebras,and discusses the irreducible representation of restricted Lie algebra gl₂.In Chapter 3,the definition of modular Lie superalgebra （?）（2）is given,and the structure of induced module of modular Lie superalgebra（?）（2）is determined.The irreducibility of induced modules is discussed,and all the simple modules of（?）（2）are isomorphic to the irreducible quotient modules of their induced modules.Meanwhile,the dimension of the irreducible quotient modules is obtained,and it is proved that （?）（2）also satisfies the Kac-Weisfeiler conjection.Chapter 4 computes the first cohomology of （?）（2）induced modules.