On the cohomology of restricted Heisenberg Lie algebras

Abstract

We show that the Heisenberg Lie algebras over a field $F$ of characteristic $p>0$ admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 1- and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and ${\cdot }^{\left[p\right]}$-operators.