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Section 2 Preliminaries

The contents of this section can, up to some modifications, be found in Section 3-4 of [23]. Nonetheless, we give here a self-contained presentation to better fit our constructive approach.

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 [14] 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.

Definition 2.1.

A Lie algebra \(\mathfrak{g}\) with base field \(F'\) is defined over a field \(F\subset F'\) if it has a basis such that the structure coefficients in that basis are all elements of \(F\text{.}\)

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 3.7 or by an immediate argument as for maximal gradings in Remark 3.12.

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. 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.

Definition 2.2.

A grading of a Lie algebra \(\mathfrak{g}\) is a direct sum decomposition \(\mathcal V : \mathfrak{g}=\bigoplus_{\alpha\in A}V_\alpha\text{,}\) where \(A\) is an abelian group and for each \(\alpha,\beta\in A\) it holds \([V_\alpha,V_\beta]\subset V_{\alpha+\beta}\text{.}\) The group \(A\) is called the grading group of the grading \(\mathcal{V}\text{,}\) and we say that the grading \(\mathcal{V}\) is over \(A\), or that \(\mathcal{V}\) is an \(A\)-grading.

The subspaces \(V_\alpha\) are called the layers of the grading \(\mathcal{V}\) and the elements \(\alpha \in A\) such that \(V_\alpha\neq 0\) are called the weights of \(\mathcal V\text{.}\) We will usually denote the set of weights by \(\Omega\text{.}\) A basis of \(\mathfrak g\) is said to be adapted to \(\mathcal V\) if every element of the basis is contained in some layer of \(\mathcal V\text{.}\)

Definition 2.4.

Let \(\mathcal V : \mathfrak g = \bigoplus_{\alpha \in A}V_\alpha \) be a grading over some abelian group \(A\text{.}\) Given an automorphism \(\Phi \in \Aut(\mathfrak g )\text{,}\) an abelian group \(B \) and a homomorphism \(f \colon A \to B \text{,}\) we define the push-forward grading \(f_* \Phi(\mathcal V) : \mathfrak g = \bigoplus_{\beta \in B}W_\beta \) over \(B\text{,}\) where

\begin{equation*} W_\beta= \bigoplus_{\alpha \in f^{-1}(\beta)} \Phi(V_{\alpha}). \end{equation*}

When \(\Phi=\operatorname{Id}\text{,}\) we simply denote \(f_* \operatorname{Id} (\mathcal{V}) = f_* \mathcal{V}\text{.}\)

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 [23]. For brevity, we will refer to this notion as equivalence. Stated in terms of push-forwards, the group-equivalence notion of [23] takes the following form:

Definition 2.5.

An \(A\)-grading \(\mathcal V\) and a \(B\)-grading \(\mathcal W\) are said to be equivalent if there exist an automorphism \(\Phi \in \Aut(\mathfrak g)\) and a group isomorphism \(f \colon A \to B \) such that \(\mathcal W = f_* \Phi(\mathcal V)\text{.}\)

In the following lemma we demonstrate that, after possibly shrinking the grading groups, an \(A\)-grading and a \(B\)-grading are equivalent if and only if they are push-forwards of each other.

Let us denote by \(\Omega_A\) and \(\Omega_B\) the sets of weights of \(\mathcal V\) and \(\mathcal W\text{.}\) Notice first that by definition of the push-forward, \(f(\Omega_A) = \Omega_B\) and \(g(\Omega_B) = \Omega_A\text{,}\) so both \(f\) and \(g\) are injective on weights. Moreover, we have for every \(\alpha \in \Omega_A\) and \(\beta \in \Omega_B\) the correspondence

\begin{equation*} V_\alpha = W_{f(\alpha)} = V_{g(f(\alpha))} \qquad \text{and} \qquad W_\beta = V_{g(\beta)} = W_{f(g(\beta))}. \end{equation*}

Hence \(f\colon \Omega_A \to \Omega_B \) is a bijection and \(f^{-1} = g\) on \(\Omega_B\text{.}\) Since \(\Omega_A\) and \(\Omega_B\) generate \(A \) and \(B\) as groups, we get that \(f^{-1} = g\) on whole \(B\text{.}\)

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 [28] and Section 1.1 of [14]. 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.

Definition 2.7.

Let \(\mathcal V : \mathfrak{g}=\bigoplus_{\alpha\in A}V_\alpha\) be a grading with weights \(\Omega \subset A \text{.}\) Suppose we have an embedding \(f \colon \Omega \to B\) into an abelian group \(B\) such that \(\mathcal{W}: \mathfrak{g}=\bigoplus_{\beta\in B}W_\beta\) is a grading, where \(W_{f(\alpha)}=V_\alpha\text{.}\) Then the resulting \(B\)-grading \(\mathcal{W}\) is called a realization of the grading \(\mathcal V\text{.}\)

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 [23].

Definition 2.8.

Let \(\mathcal V \) be a grading of \(\mathfrak g\text{.}\) A universal realization of \(\mathcal V\) is a realization \(\widetilde {\mathcal V}\) as an \(A\)-grading such that for every realization of \(\mathcal{V}\) as a \(B\)-grading, there exists a unique homomorphism \(f \colon A \to B\) such that the \(B\)-grading is the push-forward grading \(f_*\widetilde {\mathcal V}\text{.}\)

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

The universal realization can always be constructed by considering the free abelian group generated by the weights and quotienting out the grading relations, as described by the following algorithm.

In this paper, the notation \(\langle X\rangle\) always refers to the span of \(X\) in the appropriate sense.

Let \(A\) be the grading group of \(\mathcal{V}\) and consider the homomorphism \(\phi \colon \mathbb{Z}^n \to A\) defined by \(\phi(e_i)=\alpha_i\) for all \(1 \le i \le n\text{.}\) Observe that by construction \(R \subset \ker(\phi)\text{.}\) Then the grading \(\widetilde {\mathcal V}\) is well-defined: if \(\pi(e_i)=\pi(e_j)\text{,}\) then \(e_i-e_j \in \langle R\rangle\) and we have \(\alpha_i=\phi(e_i)=\phi(e_j)=\alpha_j\text{.}\) 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.

Implementation wise, step 3 contains the reduction of a matrix to its Smith Normal Form. For an estimate on its computational efficiency, we refer to Theorem 12 of [32].

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.

Let \(\widetilde{\mathcal{V}}\) be the universal realization of \(\mathcal{V}\text{.}\) By Algorithm 2.9, \(\widetilde{\mathcal{V}}\) is a \(\mathbb{Z}^n/\langle R\rangle\)-grading for some subset \(R\subset \mathbb{Z}^n\text{.}\) The quotient \(\mathbb{Z}^n/\langle R\rangle\) is isomorphic to a group \(\mathbb{Z}^k \times G_t\text{,}\) where \(G_t\) is some torsion group.

By assumption the grading group \(A\) of \(\mathcal{V}\) is torsion-free. Since the image of \(G_t\) under a homomorphism must vanish in \(A\text{,}\) we conclude that there are no non-zero weights in \(G_t\text{.}\) Since the grading group of a universal realization is generated by the weights, we conclude that \(G_t=0\text{,}\) and \(\widetilde{\mathcal{V}}\) is a \(\mathbb{Z}^k\)-grading.

The following lemma is a part of Proposition 3.15 of [23], 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 [23]. 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 [23], 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 [16].

Definition 2.12.

An abelian subalgebra \(\mathfrak{t}\) of semisimple derivations of \(\mathfrak{g}\) is called a torus of \(\der(\mathfrak g)\text{.}\) If the torus \(\mathfrak{t}\) is diagonalizable over the base field of \(\mathfrak{g}\text{,}\) it is called a split torus.

Let \(X_1,\ldots,X_n\) be a basis of \(\mathfrak{g}\) that diagonalizes \(\mathfrak{t}\text{.}\) Since each vector \(X_i\) is an eigenvector of every derivation \(\delta\in\mathfrak{t}\text{,}\) there are well defined linear maps \(\alpha_1,\ldots,\alpha_n\in\mathfrak{t}^*\) determined by

\begin{equation*} \delta(X_i) = \alpha_i(\delta)X_i,\quad i=1,\ldots,n\text{.} \end{equation*}

By construction \(X_i\in V_{\alpha_i}\text{,}\) so the direct sum \(\bigoplus_{\alpha\in\mathfrak{t}^*}V_\alpha\) spans all of the Lie algebra \(\mathfrak{g}\text{.}\) The inclusion \([V_\alpha,V_\beta]\subset V_{\alpha+\beta}\) follows by linearity from the Leibniz rule \(\delta([X,Y]) = [\delta(X),Y]+[X,\delta(Y)]\) for all derivations \(\delta\in\mathfrak{t}\) and vectors \(X\in V_\alpha\) and \(Y\in V_\beta\text{.}\)

For the purposes of Subsection 2.5, we need the following two lemmas. In Lemma 2.15 we link equivalences and push-forwards of gradings to relations between the inducing tori.

To show i, suppose that \(\Ad_{\Phi}\mathfrak{t}_1=\Phi\circ\mathfrak{t}_1\circ\Phi^{-1}=\mathfrak{t}_2\) for some automorphism \(\Phi\in\Aut(\mathfrak{g})\text{.}\) Let \(g\colon\mathfrak{t}_1^*\to\mathfrak{t}_2^*\) be the linear isomorphism \(g=\Ad_{\Phi^{-1}}^*\) given by \(g(\alpha)(\delta) = \alpha(\Phi^{-1}\circ\delta\circ\Phi)\text{.}\) Then

\begin{align*} \Phi(V_\alpha) \amp= \{\Phi(X): \delta(X)=\alpha(\delta)X\,\forall \delta\in\mathfrak{t}_1 \}\\ \amp= \{Y: \Phi\circ\delta\circ\Phi^{-1}(Y)=\alpha(\delta)Y\,\forall \delta\in\mathfrak{t}_1 \}\\ \amp= \{Y: \eta(Y)=g(\alpha)(\eta)Y\,\forall \eta\in\mathfrak{t}_2 \} = W_{g(\alpha)}\text{.} \end{align*}

Hence the gradings \(\mathcal{V}\) and \(\mathcal{W}\) are equivalent, as claimed.

Regarding ii, suppose that \(\mathfrak t_1 \subset \mathfrak t_2\text{.}\) We claim that \(\mathcal{V}=f_*\mathcal{W}\) through the restriction map \(f\colon \mathfrak{t}_2^*\to \mathfrak{t}_1^*\text{,}\) \(f(\beta)=\restr{\beta}{\mathfrak{t}_1}\text{.}\) Indeed, fix a basis \(X_1,\ldots,X_n\) of \(\mathfrak{g}\) that diagonalizes the split torus \(\mathfrak{t}_2\) (and hence also the subtorus \(\mathfrak{t}_1\)). Let \(\beta_1,\ldots,\beta_n\in\mathfrak{t}_2^*\) be the maps defined by \(\delta(X_i)=\beta_i(\delta)X_i\) and define \(\alpha_i = \beta_i|_{\mathfrak{t_1}}\text{.}\) By construction \(X_i\in W_{\beta_i}\text{,}\) \(X_i\in V_{\alpha_i}\text{,}\) and \(f(\beta_i)=\alpha_i\text{,}\) proving that \(\mathcal{V}=f_*\mathcal{W}\text{.}\)

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

Let \(A\) be the torsion-free abelian grading group of the grading \(\mathcal{V}\colon \mathfrak{g}=\bigoplus_{\alpha\in A}V_\alpha\) and let \(A^*\) be the space of homomorphisms \(A\to F\text{,}\) where \(F\) is the base field of \(\mathfrak{g}\text{.}\) By reducing to the subgroup generated by the weights, we may assume \(A \) is isomorphic to \(\mathbb{Z}^m\) for some \(m \ge 1\text{.}\) For each \(\varphi\in A^*\) define the linear map

\begin{equation*} \delta_\varphi\colon\mathfrak{g}\to\mathfrak{g},\quad \delta_\varphi(X) = \varphi(\alpha)X\quad\forall X\in V_\alpha\text{.} \end{equation*}

We claim that \(\mathfrak t = \{\delta_\varphi : \varphi \in A^*\} \) is a split torus that induces a realization for \(\mathcal V\text{.}\) Indeed, a direct computation shows that all the maps \(\delta_\varphi\) are derivations. They are diagonalizable since by construction they are multiples of the identity on each layer \(V_\alpha\text{.}\) Hence \(\mathfrak t\) is a split torus.

Let then \(\mathcal W :\mathfrak{g}=\bigoplus_{\beta\in \mathfrak t^*}W_\beta\) be the \(\mathfrak{t}^*\)-grading induced by \(\mathfrak t\text{.}\) Denote by \(\Omega\) the set of weights of \(\mathcal V\text{,}\) and define a map \(f \colon \Omega \to \mathfrak t^*\) by \(f(\alpha)(\delta_\varphi) = \varphi(\alpha)\text{.}\) Then \(f\) is well-defined: if \(\varphi, \phi \in A^*\) are such that \(\delta_\varphi = \delta_\phi\text{,}\) then by the definition of \(\mathfrak t\) we have \(\varphi(\alpha) = \phi(\alpha)\) for all weights \(\alpha \in \Omega\text{.}\)

First, we show that \(V_\alpha\subset W_{f(\alpha)}\) for every \(\alpha \in A\text{.}\) By the construction of the torus \(\mathfrak t\text{,}\) for each \(X\in V_\alpha\) we have that

\begin{equation*} \delta_\varphi(X) = \varphi(\alpha)X = f(\alpha)(\delta_\varphi)X \quad \forall \delta_\varphi \in \mathfrak t. \end{equation*}

By the definition of the grading \(\mathcal W\text{,}\) we then have \(X\in W_{f(\alpha)}\) and so \(V_\alpha\subset W_{f(\alpha)}\text{.}\)

Next, we show that the map \(f\) is injective, which would prove that \(V_\alpha = W_{f(\alpha)}\) for all \(\alpha\in \Omega\) and so \(\mathcal W\) would be a realization of \(\mathcal V \text{,}\) as claimed. Note that since \(A \) is isomorphic to \(\mathbb{Z}^m\text{,}\) for every non-zero \(\alpha \in A\) there exists a homomorphism \(\varphi \in A^*\) such that \(\varphi(\alpha)\neq 0\text{.}\) Therefore, if \(\alpha, \alpha' \in \Omega\) are such that \(f(\alpha) = f(\alpha')\text{,}\) then by the construction of the map \(f\) we have

\begin{equation*} \varphi(\alpha-\alpha') = \varphi(\alpha)-\varphi(\alpha') = f(\alpha)(\delta_\varphi)-f(\alpha')(\delta_\varphi) = 0 \end{equation*}

for every homomorphism \(\varphi\colon A\to F\text{.}\) So \(\alpha = \alpha'\) and \(f\) is injective, proving that \(\mathcal W\) is a realization of \(\mathcal V \text{.}\)

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 2.19). 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.

Remark 2.18.

The maximal grading of a Lie algebra is unique up to equivalence, since maximal split tori are all conjugate (see for instance, Theorem 15.2.6. of [31]). Indeed, by Lemma 2.15i the conjugacy implies that any two maximal split tori induce equivalent gradings, so also their universal realizations are equivalent.

Let \(\mathcal{V}'\) be the realization of \(\mathcal V\) as a \(\mathfrak{t}^*\)-grading induced by a split torus \(\mathfrak{t}\) given by Lemma 2.16. Let \(\mathfrak{t}'\supset \mathfrak t \) be a maximal split torus in \(\der(\mathfrak{g})\) with induced grading \(\mathcal W'\text{.}\) By Lemma 2.15.ii, the grading \(\mathcal{V'}\) is a push-forward of \(\mathcal W'\text{.}\)

Since the maximal grading is unique up to equivalence by Remark 2.18, we may assume that \(\mathcal W\) is the universal realization of \(\mathcal W'\text{.}\) Therefore, since \(\mathcal V\) is a realization of \(\mathcal V'\text{,}\) by Lemma 2.11 the grading \(\mathcal{V}\) is a push-forward of \(\mathcal{W}\text{.}\)

Remark 2.20.

It follows from Proposition 2.19 and the discussion in Section 3.6 of [23] that maximal gradings are universal realizations of fine gradings. In Definition 3.18 of [6], maximal gradings are defined as the gradings induced by maximal split tori in the automorphism group \(\Aut(\mathfrak{g})\text{.}\) Proposition 3.20 of [6] states that maximal gradings in the sense of [6] have a universal property equivalent to Proposition 2.19, so by Lemma 2.6 any such grading is maximal also in the sense of Definition 2.17. The maximal gradings considered in [16] are the gradings induced by maximal split tori.