## Section 2 Gradings

The contents of this section can, up to some modifications, be found in Section 3-4 of [22]. We nonetheless give here a self-contained presentation to better fit our constructive approach.

### Subsection 2.1 Gradings and equivalences

In this section we define some key notions related to gradings of Lie algebras, including equivalence, push-forwards and coarsenings. We also make a distinction between two different notions of grading, with the difference being whether the indexing plays a role or not.

###### Definition 2.1.

A grading of a Lie algebra \(\mathfrak{g}\) is a direct sum decomposition \(\mathcal V : \mathfrak{g}=\bigoplus_{\alpha\in S}V_\alpha\) such that for each \(\alpha,\beta\in S\) either \([V_\alpha,V_\beta]=0\) or there exists a unique \(\gamma\in S\) such that \([V_\alpha,V_\beta]\subset V_{\gamma}\text{.}\) When \(S\) is an abelian group \(A\) such that the unique element \(\gamma\) is given by \(\gamma=a+b\text{,}\) we say that the grading \(\mathcal{V}\) is over \(A\), or that \(\mathcal{V}\) is an \(A\)-grading. In this case, \(A\) is the grading group of the grading \(\mathcal{V}\text{.}\)

The subspaces \(V_\alpha\) are called the layers of the grading \(\mathcal{V}\) and the elements \(\alpha \in S\) 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 2.2.

Suppose the indexing set \(S\) of a grading \(\mathcal V : \mathfrak{g}=\bigoplus_{\alpha\in S}V_\alpha\) can be embedded into an abelian group \(A\) such that \([V_\alpha,V_\beta]\subset V_{\alpha+\beta}\) for all \(\alpha,\beta\in A\text{,}\) where we define \(V_\alpha=0\) for \(\alpha\notin S\text{.}\) Then the resulting \(A\)-grading is called a realization of the grading \(\mathcal V\text{.}\)

###### Definition 2.3.

A grading is called torsion-free if it admits a realization over a torsion-free (abelian) group.

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

###### Example 2.4.

Consider the 6-dimensional Lie algebra spanned by the vectors \(X_1,Y_1,Z_1,X_2,Y_2,Z_2 \) with the non-trivial bracket relations

The subspace decomposition

defines a grading. It can realized over \(\mathbb{Z}^2\) with the embedding

###### Definition 2.5.

Let \(\mathcal V : \mathfrak g = \bigoplus_{\alpha \in A}V_\alpha \) be an \(A\)-grading for 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

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 2.1.

###### Definition 2.6.

Let \(\mathfrak{g}\) be a Lie algebra and let \(\mathcal V : \mathfrak{g}=\bigoplus_{\alpha\in S_1}V_\alpha\) and \(\mathcal W: \mathfrak{g}=\bigoplus_{\beta\in S_2}W_\beta\) be two gradings. If for every \(\alpha\in S_1\) there exists \(\beta\in S_2\) such that \(V_\alpha\subset W_\beta\text{,}\) then we say that \(\mathcal V\) is a refinement of \(\mathcal W\text{,}\) and that \(\mathcal W\) is a coarsening of \(\mathcal V\text{.}\)

###### Remark 2.7.

If \(\mathcal W = {f}_*\mathcal V\) for some homomorphism \(f\text{,}\) then \(\mathcal W \) is a coarsening of \(\mathcal V\text{.}\) Such a map \(f\) is injective on the weights if and only if \(\mathcal V\) and \(\mathcal W\) are realizations of the same grading.

There are several different notions of equivalence of gradings in the literature. The two that we shall use are distinguished as equivalence and group-equivalence in [22]. For brevity, we will refer to both notions as equivalence. Stated in terms of push-forwards, the group-equivalence notion of [22] takes the following form:

###### Definition 2.8.

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 \(\mathcal W = f_* \Phi(\mathcal V)\text{.}\)

For gradings that admit realizations, the equivalence notion of [22] can be rephrased through the previous notion as follows.

###### Definition 2.9.

A grading \(\mathfrak{g}=\bigoplus_{\alpha\in S_1}V_\alpha\) and a grading \(\mathfrak{g}=\bigoplus_{\beta\in S_2}W_\beta\) over arbitrary indexing sets \(S_1,S_2\) are said to be equivalent if they admit realizations as an \(A\)-grading and a \(B\)-grading that are equivalent in the sense of Definition 2.8.

###### Example 2.10.

Consider two gradings \(V_1 \oplus V_2\) and \(V_1\oplus V_3\) of \(\mathbb{R}^2\) over \(\mathbb{Z}\) with the same one-dimensional layers. The two gradings are equivalent in the sense of Definition 2.9, since the former is a realization of the second by the embedding \(\{1,3\}\hookrightarrow\{1,2\}\subset\mathbb{Z}\text{,}\) but they are not equivalent as \(\mathbb{Z}\)-gradings in the sense of Definition 2.8 as there does not exist an automorphism of \(\mathbb{Z}\) mapping \(\{1,3\} \to \{1,2\}\text{.}\)

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.

###### Lemma 2.11.

Let \(\mathcal V : \mathfrak g = \bigoplus_{\alpha \in A}V_\alpha \) be an \(A\)-grading and \(\mathcal W : \mathfrak g = \bigoplus_{\beta \in B}W_\beta \) be a \(B\)-grading such that the weights of \(\mathcal V \) and \(\mathcal W\) generate the abelian groups \(A\) and \(B\text{,}\) respectively. If there exist homomorphisms \(f \colon A \to B \) and \(g\colon B \to A\) such that \(\mathcal W = f_*\mathcal V\) and \(\mathcal V = g_*\mathcal W\text{,}\) then \(\mathcal V\) and \(\mathcal W\) are equivalent.

###### Proof.

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

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 that the weights generate 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 2.2 Universal gradings

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, see for instance Example 3.13. To build a satisfactory theory using homomorphisms between grading groups, we consider the notion of an (abelian) universal realization, see Section 3.3 of [22].

###### Definition 2.12.

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 with \(B\) abelian, 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 2.11, the universal realization of a grading is unique up to equivalence.

If a grading admits a realization, then it also admits a universal realization. The universal realization can be constructed by considering the free abelian group generated by the weights and quotienting out the grading relations, as described by the following algorithm.

###### Algorithm 2.13. Universal realization.

Input: a grading \(\mathcal V\) that has a realization. Output: a universal realization \(\widetilde {\mathcal V}\) of \(\mathcal V \text{.}\)

- Let \(\Omega = \{\alpha_1,\dots,\alpha_n \}\) be the set of weights of \(\mathcal V\) and let \(B = \{e_1,\dots,e_n \}\) be the canonical basis of \(\mathbb{Z}^n\text{.}\) Set \(R = \emptyset\text{.}\)
- Repeat for all pairs \(\alpha_i, \alpha_j \in \Omega\text{:}\) If \(0 \neq [V_{\alpha_i}, V_{\alpha_j}] \subseteq V_{\alpha_k} \) for some \(\alpha_k \in \Omega\text{,}\) extend \(R\) by \(e_i + e_j - e_k \text{.}\)
- 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. Return the obtained \(\mathbb{Z}^n/\langle R\rangle\)-grading.

###### Proof of correctness.

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.

###### Lemma 2.14.

If \(\mathcal V\) is a torsion-free grading, then the grading group of the universal realization of \(\mathcal V\) is isomorphic to some \(\mathbb{Z}^k\text{.}\)

###### Proof.

Let \(\widetilde{\mathcal{V}}\) be the universal realization of \(\mathcal{V}\text{.}\) By Algorithm 2.13, \(\widetilde{\mathcal{V}}\) 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 there exists a realization of \(\mathcal{V}\) as an \(A\)-grading with \(A\) 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 a universal realization is generated by its 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 [22], and we record it for later usage.

###### Lemma 2.15.

If a grading \(\mathcal V\) is a coarsening of a grading \(\mathcal W \text{,}\) then every realization of \(\mathcal V\) is a push-forward grading of the universal realization of \(\mathcal W\text{.}\)

### Subsection 2.3 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 [22]. 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 [22], 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 3.3.

We start by defining split tori and gradings induced by them in the sense of [15].

###### Definition 2.16.

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.

###### Lemma 2.17.

Let \(\mathfrak t\) be a split torus of \(\der(\mathfrak g)\) and let \(\mathfrak{t}^*\) be its dual as a vector space. For each \(\alpha\in\mathfrak{t}^*\) define the subspace

Then \(\mathfrak{g}=\bigoplus_{\alpha\in\mathfrak{t}^*}V_\alpha\) is a \(\mathfrak{t}^*\)-grading.

###### Proof.

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

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 \(Y\in V_\beta\text{.}\)

###### Definition 2.18.

The \(\mathfrak{t}^*\)-grading of \(\mathfrak{g}\) defined in Lemma 2.17 is called the grading induced by the split torus \(\mathfrak{t}\).

See Example 3.13 for some gradings induced by tori in the Heisenberg Lie algebra.

For the purposes of Subsection 2.4, we need the following two lemmas. In Lemma 2.19 we link equivalences and push-forwards of gradings to relations between the inducing tori.

###### Lemma 2.19.

Let \(\mathfrak{t}_1\) and \(\mathfrak{t}_2\) be two split tori of \(\der(\mathfrak{g})\) with respective induced \(\mathfrak{t}_1^*\)-grading \(\mathcal V\) and \(\mathfrak{t}_2^*\)-grading \(\mathcal W\text{.}\)

- If there exists an automorphism \(\Phi \in \Aut(\mathfrak{g})\) such that \(\Phi \circ \mathfrak t_1 \circ \Phi^{-1} = \mathfrak t_2\text{,}\) then \(\mathcal V\) and \(\mathcal W\) are equivalent.
- If \(\mathfrak t_1 \subset \mathfrak t_2\text{,}\) then there exists a homomorphism \(f\) so that \(\mathcal V = f_* \mathcal W\).

###### Proof.

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 \(f\colon\mathfrak{t}_1^*\to\mathfrak{t}_2^*\) be the linear isomorphism \(f=\Ad_{\Phi^{-1}}^*\) given by \(f(\alpha)(\delta) = \alpha(\Phi^{-1}\circ\delta\circ\Phi)\text{.}\) Then

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 \(\mathcal{V}=g_*\mathcal{W}\) through the restriction map \(g\colon \mathfrak{t}_2^*\to \mathfrak{t}_1^*\text{,}\) \(g(\beta)=\restr{\beta}{\mathfrak{t}_1}\text{.}\) 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 \(g(\beta_i)=\alpha_i\text{,}\) proving that \(\mathcal{V}=g_*\mathcal{W}\text{.}\)

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

###### Lemma 2.20.

Let \(\mathcal V\) be a torsion-free grading. Then there exists a split torus \(\mathfrak{t}\) whose induced \(\mathfrak{t}^*\)-grading is a realization of \(\mathcal{V}\text{.}\)

###### Proof.

Let \(\mathcal{V}\colon \mathfrak{g}=\bigoplus_{\alpha\in A}V_\alpha\) be a realization of \(\mathcal{V}\) over a torsion-free abelian group \(A\) 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

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

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

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 2.4 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 2.23). The formulation through the derivation algebra will be convenient in the construction of maximal grading in Subsection 3.3. The universal property will be exploited in Subsection 2.5 where we give a method to construct all gradings over torsion-free abelian groups of a Lie algebra from a given maximal grading.

###### Definition 2.21.

Let \(\mathfrak g\) be a Lie algebra. A maximal grading of \(\mathfrak g\) is the universal realization of the grading induced by a maximal split torus of \(\der(\mathfrak g)\text{.}\)

###### Remark 2.22.

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 [29]). Indeed, by Lemma 2.19i the conjugacy implies that any two maximal split tori induce equivalent gradings, so also their universal realizations are equivalent.

###### Proposition 2.23.

Let \(\mathcal W\) be a maximal grading of \(\mathfrak{g}\) and \(\mathcal{V}\) a grading of \(\mathfrak{g}\text{.}\) Then every torsion-free realization of \(\mathcal{V}\) is a push-forward of \(\mathcal W\text{.}\)

###### Proof.

Let \(\mathcal{V}'\) be the realization of \(\mathcal V\) as a \(\mathfrak{t}^*\)-grading induced by a split torus \(\mathfrak{t}\) given by Lemma 2.20. Let also \(\mathfrak{t}'\supset \mathfrak t \) be a maximal split torus in \(\der(\mathfrak{g})\) with the induced grading \(\mathcal W'\text{.}\) By Lemma 2.19.ii, the grading \(\mathcal{V'}\) is a push-forward of \(\mathcal W'\text{.}\) In particular, \(\mathcal V\) is a coarsening of \(\mathcal W'\text{.}\)

Since the maximal grading is unique up to equivalence by Remark 2.22, we may assume that \(\mathcal W\) is the universal realization of \(\mathcal W'\text{.}\) Therefore, by Lemma 2.15 every realization of \(\mathcal{V}\) is a push-forward grading of \(\mathcal{W}\text{.}\)

###### Remark 2.24.

It follows from Proposition 2.23 and the discussion in Section 3.6 of [22] 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 2.23, so by Lemma 2.11 any such grading is maximal also in the sense of Definition 2.21. The maximal gradings considered in [15] are the gradings induced by maximal split tori.

### Subsection 2.5 Enumeration of torsion-free gradings

Following the method suggested in Section 3.7 of [22], we now give a simple way to enumerate a complete (and finite) set of universal realizations of gradings of a Lie algebra using the maximal grading. This reduces the proof of Theorem 1.4 to the construction of a maximal grading, which we cover in Subsection 3.3.

For the rest of this section, let \(\mathfrak g\) be a Lie algebra and let \(\mathcal W : \mathfrak g = \bigoplus_{n\in \mathbb{Z}^k}W_n \) be a maximal grading of \(\mathfrak g\) with weights \(\Omega\text{.}\) Denote by \(\Omega-\Omega\) the difference set \(\Omega- \Omega = \{n-m \,\mid\, n,m\in \Omega\}\text{.}\) For a subset \(I \subset \Omega-\Omega \text{,}\) let

be the canonical projection. We define the finite set

###### Proposition 2.25.

The set \(\Gamma\) is, up to equivalence, a complete set of universal realizations of torsion-free gradings of \(\mathfrak g\text{.}\)

###### Proof.

Let \(\mathcal{V}\) be the universal realization of some torsion-free grading. Due to Lemma 2.14, the grading group of \(\mathcal V\) is some \(\mathbb{Z}^m\text{.}\) By Proposition 2.23, there exists a homomorphism \(f\colon \mathbb{Z}^k\to \mathbb{Z}^m \) and an automorphism \(\Phi \in \Aut(\mathfrak g)\) such that \(\mathcal V= f_*\Phi(\mathcal W) \text{.}\) Let

We are going to show that \(\mathcal V' = (\pi_I)_*(\mathcal W) \) is equivalent to \(\mathcal{V}\text{.}\) Then, a posteriori, \(\mathbb{Z}^k/\langle I\rangle\) is torsion-free and we have \(\mathcal V' \in \Gamma\text{,}\) proving the claim.

First, since \(\ker(\pi_I) = \langle I\rangle \subseteq \ker(f)\text{,}\) by the universal property of quotients there exists a unique homomorphism \(\phi \colon \mathbb{Z}^k/\langle I\rangle \to \mathbb{Z}^m \) such that \(f = \phi \circ \pi_I\text{.}\) In particular,

so \(\mathcal V\) is a push-forward grading of \(\mathcal V'\text{.}\)

Secondly, since also \(\ker(f)\cap (\Omega-\Omega) = I \subseteq \ker(\pi_I)\cap (\Omega-\Omega) \text{,}\) we deduce that \(\mathcal V \) and \(\Phi(\mathcal V')\) are realizations of the same grading. Since \(\mathcal V\) is a universal realization, it follows that \(\Phi(\mathcal V')\) is a push-forward grading of \(\mathcal V\text{.}\) Consequently, \(\mathcal V'\) is a push-forward grading of \(\mathcal V\text{.}\) Since the grading group of a universal realization is generated by the weights, we get that the gradings \(\mathcal V\) and \(\mathcal V'\) are equivalent by Lemma 2.11, as wanted.

Notice that some of the \(\mathbb{Z}^k/\langle I\rangle\)-gradings in \(\Gamma\) are typically equivalent to each other. From the classification point of view, a more challenging task is to determine the equivalence classes once the set \(\Gamma\) is obtained. In low dimensions, naive methods are enough to separate non-equivalent gradings, and for equivalent ones the connecting automorphism can be found rather easily.

In [19] we give a representative from each equivalence class of \(\Gamma\) for every 6-dimensional nilpotent Lie algebra over \(\mathbb{C}\) and for an extensive class of 7-dimensional Lie algebras over \(\mathbb{C}\text{.}\) The results and the methods for distinguishing the equivalence classes of the obtained gradings are described in more detail in Subsection 4.2.