On completeness of some pro-solvable Lie algebras

Abstract

In the paper we describe derivations of two $\mathbb{N}$-graded infinity-dimensional Lie algebras $\mathbf{n}_1$ and $\mathbf{n}_2$ what are positive parts of affine Kac-Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$, respectively. Then we construct all pro-solvable Lie algebras whose potential nilpotent ideals are $\mathbf{n}_1$ and $\mathbf{n}_2$ and compute low-dimensional (co)homology groups of the pro-solvable Lie algebras constructed.