Generators And Automorphism Groups Of Simple Lie Superalgebras

The theory of Lie superalgebras is an important and active field of the Lie theory,which has a close relationship to theoretical physics and many branches of mathematics.Lie superalgebras over fields of characteristic p>0are also called modular ones. Inthis thesis, we mainly study the structures of characteristic zero Lie superalgebras andcharacteristic p Lie superalgebras. One work is to determine the minimal number ofgenerators for a simple Lie superalgebra and the other is to characterize the automorphismgroups of certain Lie superalgebras and the so-called Hom-operators on certain simple Liesuperalgebras. The new results obtained in this thesis are formulated as follows.First, according to the classification theorem of finite-dimensional simple Lie su-peralgebras over an algebraically closed field of characteristic zero, using the root spacedecompositions of simple Lie superalgebras and the structural properties of the local partsof Lie superalgebras of Cartan type, we prove that any finite-dimensional simple Lie su-peralgebra over an algebraically closed field of characteristic zero can be generated byone element and formulate the concrete realization of such a generator. Since an algebra-ic system is completely determined by its generators, this result can be used to study thestructures and representations of simple Lie superalgebras and can be also used to studysolvable radicals of finite-dimensional non-simple Lie superalgebras.Second, one has known eight families of finite-dimensional simple characteristicp Lie superalgebras of Cartan type. Using the fact that such a Lie superalgebra canbe generated by its highestZ-component and the1-component, we prove that a finite-dimensional simple characteristic p Lie superalgebra of Cartan type can been generatedby one or two elements; Using the simplicity and the invariance of the standard filtrationsunder the automorphism groups, we prove that there are only the trivial Hom-structureson those Lie superalgebras; Following Wilson’s method used in automorphism groups ofmodular Lie algebras, we can characterize the automorphism groups of the so-called oddHamiltonian superalgebras and their relatives and establish isomorphisms between theautomorphism group of the Lie superalgebras under consideration and the automorphis-m groups of certain associative superalgebras. In view of the fact that the classificationremains open for the finite-dimensional simple characteristic p Lie superalgebras, our re-sults can be used to further study the structures, representations and the classification of those simple Lie superalgebras.Finally, according to the classification theorem due to Kac, the infinite-dimensionalsimple linearly compact Lie superalgebras over the field of complex numbers consist often series of basic Lie superalgebras and five series of exceptional Lie superalgebras.Using the weight space decompositions and the bi-transitivity of those Lie superalgebras,we prove that the eight series ofZ-graded simple Lie superalgebras of vectorial fieldswhich are associated with the basic Lie superalgebras can be generated by one elementand, the five series ofZ-graded simple Lie superalgebras of vectorial fields which areassociated with the exceptional Lie superalgebras can been generated by two elements.This result can be used to further study the structures and representations of the infinite-dimensional simple linearly compact Lie superalgebras.