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

TL;DR

This paper investigates the structure of two families of finite-dimensional simple Lie superalgebras of Cartan type, showing they lack nondegenerate associative forms and have trivial second cohomology groups, advancing understanding of their algebraic properties.

Contribution

It demonstrates the absence of nondegenerate associative forms and the vanishing of second cohomology groups for the HO and KO Lie superalgebras, providing new insights into their cohomological structure.

Findings

Neither HO nor KO has a nondegenerate associative form.

The second cohomology groups with trivial coefficients are zero.

Superderivations into dual modules are used to establish cohomology results.

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.