Associative forms and second cohomologies of Lie superalgebras $HO$ and $KO$

Abstract

We consider two families of finite-dimensional simple Lie superalgebras of Cartan type, denoted by HO and KO, over an algebraically closed field of characteristic p>3. Using the weight space decompositions and the principal gradings we first show that neither HO nor KO possesses a nondegenerate associative form. Then, by means of computing the superderivations from the Lie superalgebras in consideration into their dual modules, the second cohomology groups with coefficients in the trivial modules are proved to be vanishing.