Noncommutative projective schemes from free nilpotent Lie algebra

In the present talk we discuss noncommutative complete projective schemes within Kapranov's model of noncommutative algebraic geometry. The projective NC-space $P_{q}?$ represents the universal enveloping (graded) algebra $S_{q}=U(g_{q}(x))$ of the free nilpotent Lie algebra $g_{q}(x)$ of index $q$ generated by $x=(x?,?,x_{n})$. We describe the NC-complete subschemes of $P_{q}?$ for $q=2$ based on differential chains in $S_{q}$. In the general case we propose the functor $B(P?,f_{q},O(-2),?,O(-q-1))$ in terms of the twisted sheaves $O(-2)$, $?$, $O(-q-1)$ on $P?$ to restore the coordinate ring of $P_{q}?$ which is reduced to $S_{q}$, and finally calculate the related cohomology groups $H^{i}(P_{q}?,O_{q}(d))$, $iЎЭ0$.