Superderivations And Maximal Subalgebras Of Cartan Type Lie Superalgebras

As a natural generalization of Lie algebras, Lie superalgebras become an efficient tool for analyzing the properties of physical systems. The theory of Lie superalgebras is closely related to many branches of mathematics. Cartan type Lie superalgebras play an important role in the category of Lie superalgebras. The present thesis is devoted to studying the superderivations and the maximal graded subalgebras of graded Cartan type Lie superalgebras.Firstly, we use a uniform method to determine the surperderivation algebras of Z-graded Cartan type modular Lie superalgebras, including the infinite dimensional or finite dimensional non-simple ones:the infinite dimensional case is reduced to the finite dimen-sional case and the latter is further reduced to the restrictedness case, which proves to be far more manageable. Certain known results on finite dimensional simple ones are also covered. In particular, the structures and the dimension formulas in the finite dimensional case are described for the outer superderivation algebras of those Lie superalgebras.Then, the derivation space from the even part of the special odd Hamilton simple modular Lie superalgebra g into the even part of the generalized Witt modular Lie su-peralgebra (?) and the odd part of the generalized Witt modular Lie superalgebra W are determined, respectively. Motivated by the work on modular Lie algebra, in this thesis we do not compute directly the derivations, but adopt a general reduction method. Our work is heavily depend on the weight space decomposition. Moreover, it is sufficient to consider the weight vectors in (?)(W) of the same weights for the generators of g.For the problems on maximal subalgebras of Lie superalgebras, we mainly research the maximal graded subalgebras of Z-graded or Zn-graded Cartan type finite dimension-al non-modular simple Lie superalgebras and Z-graded Cartan type finite dimensional restricted simple modular Lie superalgebras, respectively. Interest in the maximal sub-algebras of finite dimensional simple algebras and superalgebras is natural because of their connection with the classification problems. In this thesis, the structures of the local parts for the corresponding Lie superalgebras are studied. In particular, the structures of1-graded items as modules of0-graded items are described. Then all maximal graded subalgebras are completely determined by virtue of a constructive method and their iso-morphism classes; dimension formulas are given except for maximal S-subalgebras. In the exceptional situation, the classification of maximal S-subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie (super)algebras.