Skip to main content

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 S}V_\alpha\label{eq-grading-direct-sum}\tag{1.1} \end{equation}

indexed by some set \(S\) in such a way that for each pair \(\alpha,\beta\in S\) there exists \(\gamma\in S\) such that

\begin{equation*} [V_\alpha,V_\beta]\subset V_{\gamma}\text{.} \end{equation*}

In this paper, we will focus on Lie algebras defined over fields of characteristic zero and gradings indexed over torsion-free abelian groups, where the element \(\gamma\) is given by \(\gamma=\alpha+\beta\text{.}\)

An important example of a Lie algebra grading is the so called maximal grading (also known as fine grading), that is a grading that does not admit any proper refinement into smaller subspaces \(V_\alpha\text{.}\) A classical example of such 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 [20]. There has been a growing interest in the study of (maximal) gradings of semisimple Lie algebras since the paper [26], see the survey [22] or the monograph [13] 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 [14].

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 [15]. One of the main results in [15] is that, given a nilpotent Lie algebra \(\mathfrak{g}\) 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 \(s\) with \(r\) generators. This finiteness in the number of maximal gradings is in contrast with the existence of an uncountable number of non-isomorphic nilpotent Lie algebras 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 (also called Carnot gradings). 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 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 have played a central role in the fields of geometric analysis, geometric measure theory, and large scale geometry, see [23] for a long 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 \(G\rtimes \mathbb{R}\) [18], where \(G\) is a nilpotent Lie group and the action of \(\mathbb{R}\) on \(G\) is given by the one-parameter family of automorphisms associated with 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 [4].

Another active area of research that contains several open problems relating to positively gradable Lie groups is the quasi-isometric classification of locally compact groups. A survey on the topic can be found in [7]. 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 [8]. The quasi-isometric classification is open also for Heintze groups, see [9] for some known results.

Subsection 1.2 Main results

In all of the following statements, let \(\mathfrak{g}\) be a finite dimensional Lie algebra defined in terms of its structure coefficients and let \(F\) be the base field of \(\mathfrak{g}\text{.}\) That is, we assume we have a fixed basis \(X_1,\ldots,X_n\) of \(\mathfrak{g}\) and 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*}

Our main result is the following.

We also give explicit constructions for stratifications and positive gradings.

Theorem 1.2 and Theorem 1.3 are constructive versions of the characterizations of stratifiability in Lemma 3.10 of [6] and existence of a positive grading in Proposition 3.22 of [6].

Using Theorem 1.1, we are able to enumerate all torsion-free gradings.

A torsion-free grading is a grading that can be indexed over a torsion-free abelian group, and gradings are considered equivalent if there is an automorphism of the Lie algebra mapping layers of one grading to layers of the other. The precise definitions can be found in Section 2. The finite set we construct in Theorem 1.4 will in general contain redundant gradings, i.e., there may exist equivalent gradings in the collection. We eliminate this redundancy in the case of nilpotent Lie algebras of dimension up to 6 to find a complete classification up to equivalence of torsion-free gradings.

For applications related to positive gradings, we also give a method to extract from the complete list of Theorem 1.4 of all torsion-free gradings those that admit a positive realization, i.e., can be indexed over the positive reals:

We also give two applications of the enumeration of positive gradings obtained from the above results. First, we show that all non-equivalent positive gradings define non-isomorphic Heintze groups, see Proposition 4.10. In this way we are able to enumerate diagonal Heintze groups. Thus we give methods to tackle the problem of finding all Heintze groups with prescribed nilradical, which is a question already posed by Heintze in [18].

Second, the enumeration 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 related to gradings. The core concepts of realization, push-forward, and equivalence are defined in Subsection 2.1 and universal realizations are recalled in Subsection 2.2. Subsection 2.3 recalls how to study torsion-free gradings of a Lie algebra \(\mathfrak{g}\) in terms of subtori of the derivation algebra \(\der(\mathfrak{g})\text{.}\) Maximal gradings and their universal property are covered in Subsection 2.4. Subsection 2.5 reduces Theorem 1.4 on enumeration of gradings to proving Theorem 1.1 on algorithmic construction of a maximal grading.

In Section 3 we give the remaining constructions for our main results. Subsection 3.1 covers Theorem 1.2 on stratifiability. Subsection 3.2 covers Theorem 1.3 and Theorem 1.5 on positive gradings. An alternate approach to deciding the existence of a positive realization is also described in Appendix A. Subsection 3.3 covers Theorem 1.1 on maximal gradings.

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 dimensionality 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 enumeration of Heintze groups. Finally, in Subsection 4.4 we present the method to find improved bounds for the non-vanishing of the \(\ell^{q,p}\) cohomology.