Associative Form And Second Cohomology Of Modular Lie Superalgebra O

Let F be an algebraically closed field and charF=p>3. In this thesis, we will determine associative form and the second cohomolony group of the modular Lie super-algebra O. As is well known, if a modular Lie superalgebra L is simple and does not possess any nondegenerate associative form, then H2(L,F) and H1(L, L*) are isomorphic. Thus the second cohomology groups of modular Lie superalgebras of Cartan type could be determined by the computation of H1(L,L*). In this paper, using the weight space decompositions and Z-grading structure, we first show that modular Lie superalgebra O does not possesses a nondegenerate associative form. By means of computing the deriva-tions from the modular Lie superalgebra O in consideration into their dual modules, the second cohomology groups of O are proved to be vanishing.