On the pair of nilpotent Lie algebras

TL;DR

This paper classifies pairs of finite dimensional nilpotent Lie algebras based on a specific invariant, focusing on those where the invariant equals 6 or 7, thereby advancing understanding of their structural properties.

Contribution

It provides a classification of all such pairs with the invariant values 6 and 7, which was previously unexplored.

Findings

Complete classification for pairs with s(N,L)=6

Complete classification for pairs with s(N,L)=7

Enhanced understanding of the structure of nilpotent Lie algebra pairs

Abstract

Let $(N,L)$ be a pair of finite dimensional nilpotent Lie algebras and $N$ admits a complement $K$ in $L$ such that $\dim N=n$ and $\dim K=m$. Let $s(N,L)=\frac{1}{2}(n-1)(n-2)+1+(n-1)m-\dim \M(N,L)$, where $\M(N,L)$ denotes the multiplier of the pair $(N,L)$. The aim of this paper is to classify all pairs of finite dimensional nilpotent Lie algebras $(N,L)$, for which $s(N,L)=6,7$.