Second Cohomology Of The Modular Lie Superalgebras Of Cartan Type

The present thesis is devoted to studying the second cohomology groups of modularLie superalgebras of Cartan type. As is well known, the theories of modular Lie algebrasand Lie superalgebras over a field of characteristic zero have obtained plentiful fruits; forexample, the classifications have been settled for finite-dimensional simple Lie algebrasover a field of characteristic p > 3 and for finite-dimensional simple Lie superalgebrasover a field of characteristic zero, respectively. As a natural generalization of Lie algebras,Lie superalgebras become e?cient tools for analyzing the properties of physical systems.In the last ten years, many important results of Lie superalgebras have been obtained,but the classification problem is still open for the finite-dimensional simple modular Liesuperalgebras. Cohomology plays an important role in the research of the classificationproblem. The second cohomology groups of modular Lie algebras of Cartan type have beendetermined by Farnsteiner and Chiu. Let F be an algebracially closed field and charF =p > 2. We know that central extensions of a given Lie algebra L, or equivalently, its secondcohomology group H2(L,F), can be conveniently described by means of derivations ? :L ?→L~*, where L~* denotes the dual space of L. Farnsteiner gave a classical descriptionof central extensions H2(L,F) of Lie algebra L over prime characteristic field by means ofderivations and skew derivations. While Chiu proposed a new unifying approach, whichwas mainly based on the computation of H1(L,L~*). On the cohomology of modular Liesuperalgebras, the second cohomology groups of some simple modular Lie superalgebraspossessing nondegenerate associative forms were determined. For example, the secondcohomology groups of the finite dimensional Hamiltonian superalgebra H(m,n,t) andthe Contact superalgebra K(m,n,t) for n ? m ? 5≡0 (mod p) have been su?cientlystudied. While if a modular Lie superalgebra L is simple and does not possess anynondegenerate associative form, then H2(L,F) and H1(L,L~*) are isomorphic. Thus thesecond cohomology groups of some modular Lie superalgebras of Cartan type could be determined by the computation of H1(L,L~*). In this paper we shall determine the secondcohomology groups of the finite dimensional modular Lie superalgebras of Cartan typeW := W(m,n,t),S := S(m,n,t) and K := K(m,n,t).The paper is organized as follows. In Chapter 1, we first review brie?y the back-ground informations and some open problems on modular Lie algebras, Lie superalgebrasof characteristic zero and characteristic p > 0, respectively. In Chapter 2, we shall studythe properties of dual space derivations of finite dimensional Z-graded modular Lie super-algebras and give su?cient conditions of dual space derivations being inner. In Chapter 3and 4, we shall prove that the second cohomology groups of the finite dimensional gener-alized Witt Lie superalgebra W and special Lie superalgebra S are trivial. In Chapter 5,we shall compute the dimension of the second cohomology group of the finite dimensionalContact superalgebra K over an algebraically closed field of characteristic p > 3.