On the Cohomology of Restricted Heisenberg Lie Algebras

We show that the Heisenberg Lie algebras over an algebraically closed field F of characteristic $p>0$ admit a family of restricted Lie algebras. 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 ·[p]-operators.