  Consider the maximal nilpotent subalgebra $n_+(A_1^{(1)})$ of the simplest
affine algebra $A_1^{(1)}$ which is one of the $\mathbb{N}$-graded Lie algebras
with minimal number of generators. We show truncated versions of this algebra
in positive characteristic admit the structure of a family of restricted Lie
algebras. We compute the ordinary and restricted 1- and 2-cohomology spaces
with trivial coefficients by giving bases. With these we explicitly describe
the restricted 1-dimensional central extensions.
