Second Cohomology Group Of Modular Lie Superalgebra Ω

A class of finite-dimensional new simple modular Lie superalgebras were construct-ed in [1], over an algebraically closed fields of characteristic p>3. In this paper, we willdetermine the second cohomology group of the modular Lie superalgebras. As is wellknown, the cohomology group plays an important role in the classification of modular Liealgebras. Farnsteiner and Chiu determined the second cohomology groups of modular Liealgebras of Cartan type. A detail description of the central extensions H~2(L,F) of Liealgebra L over prime characteristic field was given by Farnsteiner by studying derivationsand skew derivations. Then the deminsions of the second cohomology groups H~2(L,F) ofthe modular Lie algebras of Cartan type were computed. But Chiu computed H~1(L, L*)directly, which gave a new approach solving the second cohomology groups of the modularLie algebras of Cartan type unifily. In the research of the cohomologies of modular Liesuperalgebras, paper [2] told us that if a modular Lie superalgebra L is simple and doesnot possess any nondegenerate associative form, then H~2(L,F) and H~1(L, L*) are isomor-phic. Second cohomology groups of some simple modular Lie superalgebras of Cartantype have been obtained. In this paper, we will determine the second cohomology groupof modular Lie superalgebras by means of computing H~1(L, L*).