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Section 1 Introduction

Subsection 1.1 Overview

A grading of a Lie algebra \(\mathfrak{g}\) is a direct sum decomposition

\begin{equation} \mathfrak{g}=\bigoplus_{\alpha\in A}V_\alpha\label{eq-grading-direct-sum}\tag{1.1} \end{equation}

indexed by an abelian group \(A\) in such a way that

\begin{equation} [V_\alpha,V_\beta]\subset V_{\alpha+\beta}\label{eq-grading-relations}\tag{1.2} \end{equation}

for each pair \(\alpha,\beta\in A\text{.}\) In this paper, we will focus on Lie algebras over fields of characteristic zero and torsion-free gradings, i.e., gradings indexed over torsion-free abelian groups. There are also more general notions of gradings of Lie algebras or more general algebras, see for instance [26] or [4].

An important example of a Lie algebra grading is the so called maximal grading, which has the property of not admitting any proper refinement into smaller subspaces \(V_\alpha\text{.}\) Slightly different notions of a maximal grading (sometimes also fine grading) exist in the literature. In this paper, we follow a similar viewpoint as in [14], and define maximal gradings by using maximal split tori of the derivation algebra \(\der(\mathfrak g)\text{,}\) see Definition 2.16. The subspaces \(V_\alpha\) of a maximal grading are then the maximal subspaces of \(\mathfrak{g}\) where every derivation of the torus acts by scaling. In addition, the indexing group of a maximal grading is some \(\mathbb{Z}^k\) with the universal property that every other torsion-free grading can be obtained using projections, see Proposition 2.18.

A classical example of a maximal grading is the Cartan decomposition, which plays a fundamental role in representation theory and the classification of semisimple Lie algebras over \(\mathbb{C}\text{,}\) see for example [19]. There has been a growing interest in the study of (maximal) gradings of semisimple Lie algebras since the paper [26], see the survey [21] or the monograph [12] for an overview. Moreover, a classification of maximal gradings of simple classical Lie algebras over algebraically closed fields of characteristic zero can be found in [13].

Regarding nilpotent Lie algebras over algebraically closed fields of characteristic zero, an in depth study of maximal gradings over torsion-free abelian groups was carried out in [14]. One of the main results in [14] is that within the family of nilpotent Lie algebras \(\mathfrak{g}\) of nilpotency step \(s\) and with abelianization \(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]\) of dimension \(r\text{,}\) there are only finitely many torsion-free maximal gradings, up to automorphisms of the free nilpotent Lie algebra of step \(s\) and rank \(r\text{.}\) This finiteness in the number of maximal gradings is in contrast with for example the existence of an uncountable number of non-isomorphic nilpotent Lie algebras over the reals in dimension 7 and higher.

There are two other special types of gradings of particular interest in the case of nilpotent Lie algebras: positive gradings and stratifications. A positive grading is a grading indexed over the reals such that in the direct sum decomposition (1.1) all the non-zero spaces \(V_\alpha\) have positive indices \(\alpha\gt 0\text{.}\) A stratification (also called a Carnot grading) is a positive grading for which \(V_1\) generates \(\mathfrak{g}\) as a Lie algebra.

Lie algebras with a stratification are the Lie algebras of Carnot groups. These groups play a central role in the fields of geometric analysis, geometric measure theory, and large scale geometry, see [22] for a list of references.

Positive gradings are important within the study of homogeneous spaces, as they appear directly in characterizations of such spaces. First, any negatively curved homogeneous Riemannian manifold is a Heintze group \(G\rtimes \mathbb{R}\) [17], where \(G\) is a nilpotent Lie group and the action of \(\mathbb{R}\) on \(G\) is given by a one-parameter family of automorphisms inducing a positive grading of \(G\text{.}\) Second, any connected locally compact group that admits a contracting automorphism is a positively gradable Lie group [28]. In this latter result, the group structure and contracting automorphism may also be replaced by a metric structure and a dilation, see [3].

Another active area of research that contains several open problems related to positively gradable Lie groups is the quasi-isometric classification of locally compact groups. A survey on the topic can be found in [5]. For instance, it is not known whether there exists a non-stratifiable positively gradable Lie group that is quasi-isometric to its asymptotic cone, nor whether all large-scale contractible groups are positively gradable, see Question 7.9 of [6]. The quasi-isometric classification is open also for Heintze groups, see [7] for some known results.

Subsection 1.2 Implemented algorithms and applications

The purpose of our work is to implement algorithms for computing various gradings of a Lie algebra. In this paper we give a thorough explanation of the algorithms together with relevant mathematical concepts and proofs of correctness, as well as present some applications. The source code for the implementation can be found in [18], along with a full classification of gradings in dimension 6 and for “most” Lie algebras in dimension 7, in the sense explained in Subsection 4.2.

In all of the algorithms, we work with a finite dimensional Lie algebra \(\mathfrak{g}\) defined by its structure coefficients in some field \(F\text{,}\) which does not have to be the base field of the Lie algebra and will usually be smaller. That is, we assume we have a fixed basis \(X_1,\ldots,X_n\) of \(\mathfrak{g}\) and are given a family of coefficients \(\{c_{ij}^k\in F: i,j,k\in\{1,\ldots,n\}\}\) such that the Lie bracket is defined as

\begin{equation*} [X_i,X_j] = \sum_{k=1}^nc_{ij}^kX_k\text{.} \end{equation*}

In practice, \(F\) will usually be either the field of rationals or algebraic numbers. The actual base field of the Lie algebra will usually not be relevant, see the discussion in Subsection 2.1.

The most important of the implemented algorithms is the construction of a maximal grading of a Lie algebra \(\mathfrak{g}\) when the field \(F\) is algebraically closed, see Algorithm 3.10.

Given a maximal grading of a Lie algebra \(\mathfrak{g}\text{,}\) it is straightforward to construct a finite collection of gradings, which contains the universal realizations (see Definition 2.8) of all torsion-free gradings of \(\mathfrak{g}\) up to equivalence in the sense of Definition 2.5. We cover this enumeration procedure in Subsection 3.1. The finite collection will in general contain equivalent gradings. In the classification of gradings given in [18] we eliminate this redundancy in the case of nilpotent Lie algebras of dimension up to 6.

A maximal grading of a Lie algebra \(\mathfrak{g}\) also determines a parametrization of all the positive gradings of \(\mathfrak{g}\) as a convex cone. We explain this in detail in Subsection 3.3 and present Algorithm 3.8 which is a method to construct a positive realization of a grading when one exists. We also give a construction for a stratification (when one exists) in Subsection 3.2. The stratifiability criterion is the same as the one in Lemma 3.10 of [4], which we write out as an explicit linear system.

Finally we give two applications of the parametrization of positive gradings. In Proposition 4.10 we show that all non-equivalent positive gradings define non-isomorphic Heintze groups. Thus we give methods to enumerate diagonal Heintze groups. Consequently, we make progress on the problem of finding all Heintze groups with prescribed nilradical, which is a question already hinted by Heintze, see the discussion after Theorem 3 in [17].

As another application, the parametrization of positive gradings gives a method to find better estimates for the non-vanishing of the \(\ell^{q,p}\) cohomology of a nilpotent Lie group, which is a quasi-isometry invariant.

Subsection 1.3 Structure of the paper

In Section 2 we recall various definitions and terminology. In Subsection 2.1 we discuss to what extent we need to worry about the base fields of the Lie algebras appearing throughout the paper. The core concepts of realization, push-forward, and equivalence are defined in Subsection 2.2 and universal realizations are recalled in Subsection 2.3. Subsection 2.4 covers how to study torsion-free gradings of a Lie algebra \(\mathfrak{g}\) in terms of tori of the derivation algebra \(\der(\mathfrak{g})\text{.}\) Maximal gradings and their universal property are discussed in Subsection 2.5.

In Section 3 we give our main constructions. Subsection 3.1 reduces the enumeration problem of all torsion-free gradings to the construction of a maximal grading. Subsection 3.2 covers the construction of a stratification. Subsection 3.3 considers Algorithm 3.8 on positive gradings. The construction of a maximal grading, Algorithm 3.10, is described in Subsection 3.4.

In Section 4 we give various applications of gradings to the study of Lie algebras and Lie groups. Subsection 4.1 shows how to use the maximal grading of a Lie algebra as a tool to detect decomposability of a Lie algebra, and how to reduce the dimension of the problem of deciding whether two Lie algebras are isomorphic. In Subsection 4.2 we classify up to equivalence the gradings of low dimensional nilpotent Lie algebras over \(\mathbb{C}\text{.}\) In Subsection 4.3 we cover the results on the enumeration of Heintze groups. Finally, in Subsection 4.4, we present the method for finding improved bounds for the non-vanishing of the \(\ell^{q,p}\) cohomology.