Restricted One-dimensional Central Extensions of the Restricted Filiform Lie Algebras ${\frak m}_0^\lambda(p)$

We show, for a field ${mathbb F}$ of prime characteristic $pu003e0$, that the truncated filiform Lie algebra ${frak m}_0(p)$ admits a family ${frak m}_0^lambda(p)$ of restricted Lie algebra structures parameterized by elements $lambdain {mathbb F}^p$. We compute the ordinary cohomology groups $H^q({frak m}_0^lambda(p))$ and restricted cohomology groups $H^q_*({frak m}_0^lambda(p))$ for $q=1, 2$, and we give explicit descriptions of bases for these cohomology spaces. We apply our results to restricted one-dimensional central Extensions of the algebras ${frak m}_0^lambda(p)$.