Modular Lie Superalgebras Of Cartan Type

The present thesis is devoted to studying modular Lie superalgebras (that is, Lie superalgebras over fields of prime characteristic) of Cartan type. As is well known, the theories of modular Lie algebras and Lie superalgebras of characteristic zero have seen remarkable evolutions; for instance, the classifications have been settled for finite-dimensional simple Lie algebras over algebraically closed fields of characteristic p > 7 and for finite-dimensional simple Lie superalgebras and infinite-dimensional simple linearly compact Lie superalgebras over algebraically closed fields of characteristic zero, respectively. On the other hand, the theory of modular Lie superalgebras is closely related to, but different from, those of modular Lie algebras and Lie superalgebras of characteristic zero. Therefore, the work on modular Lie superalgebras is of particular interest at present stage. In particular, the classification problem is still open for finite-dimensional simple modular Lie superalgebras. Just as in the setting of modular Lie algebras, it turns out that modular Lie superalgebras of Cartan type will play a central role in the classification of finite-dimensional simple modular Lie superalgebras. What is more, we have already found a new infinite family of finite-dimensional simple modular Lie superalgebras of Cartan type, that is, odd Hamiltonian superalgebras, which are analogous to neither finite-dimensional simple modular Lie algebras nor finite-dimensional simple Lie superalgebras of characteristic zero. This observation illustrates that the classification should be nontrivial for finite-dimensional simple modular Lie superalgebras.In Chapter 2 we first determine the automorphism groups for finite-dimensional and infinite-dimensional generalized Witt superalgebras, finite-dimensional restricted Lie superalgebras of Cartan type S, H, and K. Speaking precisely, we construct concrete embedding homomorphisms of the automorphism groups of those Lie superalgebras of Cartan type mentioned above into the automorphism groups of the corresponding underlying algebras (tensor product of the divided power algebra and exterior algebra). Since the underlying superalgebras are associative and super-commutative, we are interested in establishing those relations. In addition we determine completely the outer superderivation algebras of finite-dimensional Lie superalgebras of Cartan type W, S, H, and K; in particular, they are all abelian or metabelian Lie algebras with explicit structure.In Chapter 3 we construct a new infinite family of finite-dimensional simple modular Lie superalgebras of Cartan type, that is, odd Hamiltonian superalgebras. By establishing dimension formulas of the known modular Lie superalgebras of Cartan type, one may make a comparison between odd Hamiltonian superalgebras and the other modular Lie superalgebras of Cartan type. As a result, we find that the family of simple Lie superalgebras we construct does contain "strange" ones. Furthermore, the natural filtrationof finite-dimensional odd Hamiltonian superalgebra is proved to be invariant under the automorphism group. Using this property, we then prove that the parameters defining finite-dimensional odd Hamiltonian superalgebras are intrinsic and therefore, classify these simple Lie superalgebras in the sense of isomorphism. The automorphism group is also characterized in the sense mentioned above for restricted odd Hamiltonian superalgebras.In Chapter 4, by giving the generator set and formulating the homogeneous superderiva-tions, we determine completely the superderivation algebra and the outer superderivation algebra of the finite-dimensional odd Hamiltonian superalgebra.Chapter 5 is concerned with infinite-dimensional odd Hamiltonian superalgebras. We first prove that the natural filtration is invariant and in addition give an intrinsic property for these Lie superalgebras. Then we establish embedding homomorphisms of a family transitively Z-graded finite-dimensional or infinite-dimensional modular Lie superalgbras into infinite-dimensional odd...