Central extensions of restricted Lie superalgebras and classification of $p$-nilpotent Lie superalgebras in dimension $4$

The first main result of this paper is to build the first and second restricted cohomology groups for restricted Lie superalgebras in characteristic $p\ge 3$, modifying a construction by Yuan, Chen and Cao. We will explain how these groups capture some algebraic structures, such as extensions and derivations. Further, we apply this construction to classify $p$-nilpotent restricted Lie superalgebras up to dimension $4$ over an algebraically closed field of characteristic $p\ge 3$.