Construction of nice nilpotent Lie groups

TL;DR

This paper presents an algorithm to classify nice nilpotent Lie algebras up to dimension 9, providing complete listings and analyzing the number of inequivalent nice bases for lower dimensions.

Contribution

It introduces a classification algorithm for nice nilpotent Lie algebras and provides comprehensive listings for dimensions up to 9, including counts of inequivalent nice bases.

Findings

Complete classifications for n ≤ 9

Number of inequivalent nice bases for n ≤ 7 can be 0, 1, or 2

Any nilpotent Lie algebra has at most countably many nice bases

Abstract

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $n\leq9$. On every nilpotent Lie algebra of dimension $\leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, or $2$. We show that any nilpotent Lie algebra of dimension $n$ has at most countably many inequivalent nice bases.