Gradings for nilpotent Lie algebras

Subsection 2.1 On base fields

Throughout the paper, we will always take for granted that the base fields for all Lie algebras and other vector spaces are of characteristic zero. This characteristic zero assumption is essential in order to work interchangeably with gradings of a Lie algebra \(\mathfrak{g}\) and tori of the derivation algebra \(\der(\mathfrak{g})\text{,}\) see Section 1.4 of [12] for some discussion on the non-zero characteristic case.

At several points we will ignore the base field \(F'\) of the Lie algebra and only work with the potentially smaller field \(F\subset F'\) that the Lie algebra is defined over.

All the constructions given in Section 3 are in fact independent of the actual base field \(F'\supset F\text{.}\) In the case where the construction does not involve the base field at all this independence is automatic, such as in the enumeration of gradings in Subsection 3.1. Otherwise we will mention the reason explicitly either by a reference such as for stratifications in Remark 3.6 or by an immediate argument as for maximal gradings in Remark 3.11.

The independence of the base field is particularly convenient because it allows us to make use of computer algebra systems, where it is necessary to work with a computable field, i.e., it is necessary to be able to distinguish elements and compute all the field operations with finite algorithms. For instance, although the field of reals \(\mathbb{R}\) is not computable, we may make use of our computer algebra implementation as soon as the Lie algebra is defined over the rationals or the real algebraic numbers, i.e., the real numbers that are algebraic over the rationals. As an example, in Subsection 4.2 we will give a brief overview of our classification of gradings of all Lie algebras up to dimension 6 over \(\mathbb{C}\text{.}\)

Subsection 2.2 Gradings and equivalences

In this section we define some key notions related to gradings of Lie algebras, including push-forward and equivalence.

It is readily checked that the push-forward grading is indeed a \(B\)-grading in the sense of Definition 2.2.

There are several different notions of equivalence of gradings in the literature. The one that we shall use is called group-equivalence in [21]. For brevity, we will refer to this notion as equivalence. Stated in terms of push-forwards, the group-equivalence notion of [21] takes the following form:

The equivalence of an \(A\)-grading and a \(B\)-grading has a well known characterization in terms of push-forwards. We provide a brief proof for completeness.

Notice that the assumption on generating weights is indeed necessary: for instance, the gradings \(\mathbb{R} = V_1 \) over \(\mathbb{Z}\) and \(\mathbb{R} = V_{(1,0)} \) over \(\mathbb{Z}^2\) are push-forward gradings of each other, but they are not equivalent.

Subsection 2.3 Universal realizations

In some cases the grading in hand plays a role of a partition of the Lie algebra, where the indexing of the layers is unnatural or irrelevant, see for instance [26] and Section 1.1 of [12]. One may want to equip such a grading with a new labeling, which has more convenient algebraic structure or which reveals some special properties of the grading, like positivity. Such a reindexing is said to be another realization of the original grading.

We do not in general require that the weights of an \(A\)-grading generate the grading group \(A\) in order to include e.g. gradings over \(A=\mathbb{R}\) in the discussion. Moreover, weights of a grading may have additional relations coming from the ambient group structure, even when the corresponding layers are unrelated. To build a satisfactory theory using homomorphisms between grading groups, we consider the notion of an (abelian) universal realization, see Section 3.3 of [21].

Observe that by Lemma 2.6, the universal realization of a grading is unique up to equivalence.

The universal realization of a grading \(\mathcal{V}\) can be constructed by considering the free abelian group generated by the weights and quotienting out the grading relations as described below. In this paper, the notation \(\langle X\rangle\) always refers to the span of \(X\) in the appropriate sense.

1. Enumerate the support of \(\mathcal V\) as \(\Omega = \{\alpha_1,\dots,\alpha_n \}\) and let \(\{e_1,\dots,e_n\}\) be the canonical basis of \(\mathbb{Z}^n\text{.}\)
2. Enumerate the grading relations \(R\subset\mathbb{Z}^n\) as follows. For each pair \(\alpha_i, \alpha_j \in \Omega\) such that \([V_{\alpha_i}, V_{\alpha_j}] \neq 0\text{,}\) add the element \(e_i + e_j - e_k\) to \(R\text{,}\) where \(k\) is the index such that \(\alpha_k = \alpha_i+\alpha_j\text{.}\)
3. For all \(i=1,\dots,n\text{,}\) set \(\widetilde V_{\pi(e_i)} = V_{\alpha_i}\text{,}\) where \(\pi \colon \mathbb{Z}^n \to \mathbb{Z}^n/\langle R\rangle\) is the projection. The resulting \(\mathbb{Z}^n/\langle R\rangle\)-grading \(\widetilde {\mathcal V}\) is the universal realization.

To see that the grading \(\widetilde {\mathcal V}\) is well-defined, consider the homomorphism \(\phi \colon \mathbb{Z}^n \to A\) defined by \(\phi(e_i)=\alpha_i\) for all \(1 \le i \le n\text{,}\) where \(A\) is the grading group of \(\mathcal{V}\text{.}\) If \(\pi(e_i)=\pi(e_j)\text{,}\) then \(e_i-e_j \in \langle R\rangle\) and we have \(\alpha_i=\phi(e_i)=\phi(e_j)=\alpha_j\) since \(R\subset\ker\phi\text{.}\) Moreover, the obtained \(\mathbb{Z}^n/\langle R\rangle\)-grading is a universal realization of \(\mathcal V\) by the universal property of quotients, since the construction of \(\widetilde{\mathcal{V}}\) does not depend on the realization of \(\mathcal{V}\) that we start with.

In the rest of the paper we will focus on gradings that admit torsion-free realizations. For such gradings, the universal realizations are gradings over some \(\mathbb{Z}^k\text{,}\) as demonstrated by the following lemma.

The following lemma is a part of Proposition 3.15 of [21], and we record it for later usage.

Subsection 2.4 Gradings induced by tori

In this subsection we describe the correspondence between gradings of a Lie algebra \(\mathfrak{g}\) and the split tori of its derivation algebra \(\der(\mathfrak{g})\text{.}\) In general, gradings of a Lie algebra \(\mathfrak{g}\) are in one-to-one correspondence with algebraic quasitori, see Section 4 of [21]. However, in this study we are only interested in cases when \(\mathfrak{g}\) is a finite-dimensional Lie algebra over a field of characteristic zero and the gradings are over torsion-free abelian groups. In this setting, the characterization of gradings in terms of algebraic quasitori can be reduced to studying algebraic subtori of the derivation algebra \(\der(\mathfrak{g})\text{.}\)

For computational reasons, we will drop the algebraicity requirement for the subalgebras of \(\der(\mathfrak{g})\text{.}\) This means we lose the one-to-one correspondence described in [21], but the less restrictive definition will be sufficient for our purposes. In particular, it will simplify the explicit construction of maximal gradings in terms of tori, see Subsection 3.4.

We start by defining split tori and gradings induced by them in the sense of [14].

For the purposes of Subsection 2.5, we need the following two lemmas. In Lemma 2.14 we link equivalences and push-forwards of gradings to relations between the inducing tori.

Finally, we observe that any torsion-free grading has a realization induced by a split torus.

Subsection 2.5 Maximal gradings

We now present the notion of maximal grading using maximal split tori and prove that a maximal grading has the universal property of push-forwards (see Proposition 2.18). The formulation through the derivation algebra will be convenient in the construction of maximal grading in Subsection 3.4. The universal property will be exploited in Subsection 3.1 where we give a method to construct all gradings over torsion-free abelian groups of a Lie algebra from a given maximal grading.