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

TL;DR

This paper investigates the derivations and low-dimensional (co)homology groups of certain pro-solvable Lie algebras related to affine Kac-Moody algebra positive parts, providing explicit constructions and computations.

Contribution

It describes derivations of specific infinite-dimensional Lie algebras and constructs all pro-solvable Lie algebras with these as nilpotent ideals, including explicit (co)homology calculations.

Findings

Derivations of the Lie algebras are explicitly described.

All pro-solvable Lie algebras with given nilpotent ideals are constructed.

Low-dimensional (co)homology groups are computed for specific cases.

Abstract

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.