On derivations and low-dimensional (co)homology groups of pro-sovable Lie algebras associated with ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$

In the paper we describe the derivations of two $\mathbb{N}$-graded infinity-dimensional Lie algebras ${\mathbf{n}}_1$ and ${\mathbf{n}}_1$ what are positive parts of affine Kats-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 ${\mathbf{n}}_2$. For two specific representatives of these classes low-dimensional (co)homology groups are computed.