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 $p>0$, 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 $\lambda\in {\mathbb F}^p$. 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 ${\frak m}_0^\lambda(p)$.