The Generalized Modular Lie Superalgebras Of Cartan Type

The present thesis is devoted to studying three classes of generalized modular Liesuperalgebras of Cartan type. It is well known the complete classification of the finite-dimensional simple modular Lie superalgebras remains an open problem. So construct-ing new finite-dimensional modular Lie superalgebras of Cartan type and studying theirderivation algebra, filtration structure as well as other natural properties are of great im-portance before the problem is resolved. Prof. Zhang Yongzheng constructed four classesof finite-dimensional simple modular Lie superalgebras W,S,H,K in 1997 and provedtheir simplicity and restrictability. Based on the usually modular Lie superalgebras Wand S in literature[1], the three classes generalized modular Lie superalgebras W,W andS are constructed in our thesis. Moreover, we discuss their simplicity, restrictability andderivations and outer derivations.The thesis is organized as follows.In Chapter 1, we first review brie?y the backgrounds of modular Lie algebras andLie superalgebras over a field of characteristic zero and characteristic p > 0, respectively.Moreover, we summarize the progress on modular Lie superalgebras. Second, we introducethe arrangement of the full thesis and describe the main content of the study of thesis.In Chapter 2, we construct mainly the two classes generalized modular Lie super-algebras W and W. Based on it, we determine derivation algebras of the two classesgeneralized modular Lie superalgebras. In particular, in section 1 we construct general-ized Witt modular Lie superalgebras W and call it the first class generalized Witt modularLie superalgebras. In section 2, we study the structure of the first class generalized Wittmodular Lie superalgebras. i.e., we give a generator set of W and illustrate that W is notsimple modular Lie superalgebras. In section 3, we describe the homogeneous derivationsof Z-degree of W. Moreover, the derivation superalgebra of W is determined. In section4, we construct the second class generalized Witt modular Lie superalgebras. Comparing the first and the second class generalized Witt modular Lie superalgebras, we can obtainthat they are both relation and di?erence. In addition, we prove that the second classgeneralized Witt modular Lie superalgebras is simple in this section. It is di?erent fromthe first class generalized Witt modular Lie superalgebras. In the end of this chapter,we determine completely the derivation superalgebra of the second class generalized Wittmodular Lie superalgebra W.In Chapter 3, the so-called generalized special modular Lie superalgebra which con-tains the Lie superalgebra of Cartan type S as subalgebra is constructed. Then we provethat it is not simple Lie superalgebra and give its dimension formula. In addition, wediscuss that S is a restricted Lie superalgebra if t = 1 = (1,1,...,1)∈Nn0. Finally, thederivations and outer derivations are completely determined.In Chapter 4, we first recall some concepts on Lie triple systems and restrictedLie triple systems. Based on it, we give the necessary and su?cient conditions of thenilpotency or solvability of Hp. Second, we define quasi-toral restricted Lie triple systemsand study how a quasi-toral restricted Lie triple system T with zero center and of minimaldimension should be.