Extensions of solvable Lie algebras with naturally graded filiform nilradical

TL;DR

This paper classifies and constructs extensions of solvable Lie algebras with specific naturally graded filiform nilradicals, identifying unique non-split extensions and comparing various one-dimensional extensions.

Contribution

It provides a complete classification of one-dimensional and maximal codimension extensions of solvable Lie algebras with certain nilradicals, including the unique non-split extension.

Findings

All one-dimensional central extensions of $n_{n, 1}$ are found.

Extensions of $Q_{2n}$ are shown to be split.

A unique non-split extension of maximal codimension is identified.

Abstract

In this work we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n, 1}$ and $Q_{2n}.$ We find all one-dimensional central extensions of the algebra $n_{n, 1}$ and show that any extension of $Q_{2n}$ is split. After that we find one-dimensional extensions of solvable Lie algebras with nilradical $n_{n, 1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n, 1}$ of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical $n_{n, 1}$ whose codimension is equal to one are found and compared these solvable algebras with the solvable algebras with nilradicals are one-dimensional central extension of algebra $n_{n, 1}$.