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

Abstract

We show, for a field $\mathbb{F}$ of prime characteristic $p>0$, that the truncated filiform Lie algebra ${\mathfrak{m}}_0(p)$ admits a family ${\mathfrak{m}}_0^{\lambda }(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in {\mathbb{F}}^p$. We compute the ordinary cohomology groups $H^q({\mathfrak{m}}_0^{\lambda }(p))$ and restricted cohomology groups $H_{\ast }^q({\mathfrak{m}}_0^{\lambda }(p))$ 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 ${\mathfrak{m}}_0^{\lambda }(p)$.

16 pages