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Quaternionic Bott-Chern cohomology and existence of HKT metrics

We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [35] and Angella-Dloussky-Tomassini [3] for compact complex surfaces.


Introduction
A hypercomplex manifold (M, I, J, K) is a manifold M of real dimension 4n equipped with three complex structures I, J, K satisfying the quaternionic relations. Such a manifold admits a unique torsion-free connection preserving I, J, K called the Obata connection [27]. If the holonomy of the Obata connection lies in the commutator subgroup SL(n, H) of the general quaternionic linear group GL(n, H), then the hypercomplex manifold (M, I, J, K) is called an SL(n, H)-manifold.
From the metric point of view, we can always endow a hypercomplex manifold with a quaternionic Hermitian metric, i.e. a Riemannian metric g invariant with respect to all three complex structures I, J, K. If we impose additional restrictions on the metric, we stumble upon some well-known metrics. If, for instance, we let Ω be the (2, 0)-form with respect to I defined by Ω(·, ·) := g(J·, ·)+ √ −1 g(K·, ·), then the metric g is called hyperkähler [8] if dΩ = 0 and hyperkähler with torsion [19] (HKT for short) if ∂Ω = 0, where ∂ is the Dolbeault operator with respect to I. G. Grantcharov, M. Verbitsky and the first named author recently proved that a compact SL(2, H)-manifold is HKT if and only if the Hodge number h 0,1 (with respect to I) is even [17]. This is the quaternionic counterpart of the well-known result that a compact complex surface is Kähler if and only if the first Betti number is even [10,21,26,32]. The aim of the present paper is to further investigate this link between compact SL(2, H)-manifolds with HKT metrics on one hand and compact complex surfaces with Kähler metrics on the other hand. We adapt the following complex geometry results to the quaternionic setup: A compact complex surface (1) has its Frölicher spectral sequence degenerate at the first page [20]. (2) is Kähler if and only if the second non-Kähler-ness degree vanishes [35]. (3) is Kähler if and only if it is strongly Gauduchon [28].
A compact SL(2, H)-manifold (1) has its quaternionic Frölicher spectral sequence degenerate at the first page. The fundamental difference from the complex setup is that we are not in the presence of a double complex. Instead we are facing a single complex with two anticommuting differentials acting on the same component of the bidegree. The upshot is that most proofs simplify, but on the downside we sometimes need to impose additional assumptions, such as the SL(n, H) holonomy constraint, to recover the necessary symmetries.
The organisation of the paper is as follows. In Section 2 and 3 we recall the definition of quaternionic cohomology groups and give quaternionic versions of the Frölicher spectral sequence and of the dd c -Lemma. In Section 4 we state a Frölichertype inequality for quaternionic Bott-Chern cohomology on compact hypercomplex manifolds as proven by Angella-Tomassini in a very general setup in [5] and for compact complex manifolds in [4]. In Section 5 we take a look at quaternionic metric structures and in Section 6 we build a Hodge theory in the particular case when the holonomy of the Obata connection is contained in SL(n, H). In Section 7 we show how to obtain a type decomposition in real and imaginary (2, 0)-forms by adapting tools from almost-complex geometry [13]. In Section 8 we show that the quaternionic spectral sequence degenerates at the first page on compact SL(2, H)manifolds. In Section 9 and 10 we prove that a compact SL(2, H)-manifold admits a HKT metric if and only if it admits a quaternionic strongly Gauduchon metric, and that this happens precisely when the second non-HKT-ness degree vanishes. Finally, in Section 11 we compute the quaternionic cohomology groups of some nilmanifolds in real dimension 8 and 12.

Quaternionic cohomology groups
A hypercomplex manifold (M, I, J, K) is a smooth manifold M of real dimension 4n equipped with three integrable almost-complex structures I, J, K satisfying the quaternionic relations I 2 = J 2 = K 2 = IJK = −Id T M . We denote by i=0 Ω i (M ) the bundle of real exterior forms on M and by the decomposition of the complexified bundle with respect to I. The sections of Ω p,q I (M ) are denoted by Λ p,q I (M ). The complex structure J acts on 1-forms and this action is extended to the space of p-forms by for any p-form ϕ. Since I and J anti-commute, the action by J matches (p, q)-forms with respect to I to (q, p)-forms: . We define the twisted differential operator ∂ J : Λ p,q I (M ) → Λ p+1,q I (M ) by where ∂ is the standard Dolbeault operator with respect to I. Note that both ∂ and ∂ J increase p in the bidegree (p, q) and leave q untouched. By the integrability of I, we deduce ∂ 2 = ∂ 2 J = 0. Moreover, by a result of [37], the operators ∂ and ∂ J anti-commute: ∂∂ J + ∂ J ∂ = 0. If we fix q = 0, we get a (single) cochain complex (Λ p,0 I (M ), ∂, ∂ J ) with two anticommuting differentials. For a hypercomplex manifold (M, I, J, K) of real dimension 4n, the complex (Λ p,0 I (M ), ∂, ∂ J ) leads to the quaternionic Dolbeault cohomology groups Moreover, one can define the quaternionic Bott-Chern cohomology groups and the quaternionic Aeppli cohomology groups It has been shown in [17] that all of these groups are finite-dimensional: On a compact hypercomplex manifold, the groups H p,0 BC and H p,0 AE are finite-dimensional.
To ease the notation, we use h p,0 BC = dim H p,0 BC , h p,0 ∂J = dim H p,0 ∂J , etc. Following J. Varouchas [36], we define the complex vector spaces where ∂ and ∂ J are acting on forms of type Λ p,0 I (M ) with 0 p 2n. Once more the dimensions of these spaces will be denoted by the corresponding lower case letter, for example a p,0 = dim A p,0 .

Lemma 2. The two sequences
Proof. Exactness only relies on ∂ 2 = 0 whereas finiteness can be deduced from finiteness of h p,0 BC and h p,0 AE . As in the complex setup, Varouchas' short exact sequences will help us to quantify the discrepancy between the quaternionic Dolbeault groups and the quaternionic Bott-Chern/Aeppli groups. Last, but not least, the following map will be a crucial tool in the analysis of quaternionic cohomology groups.

Quaternionic Frölicher spectral sequence and dd c -Lemma
In this Section we introduce a quaternionic analog of the Frölicher spectral sequence and we show that degeneration at the first page is a weaker condition than the quaternionic dd c -Lemma introduced in [17]. For more information about the Frölicher spectral sequence in the complex setup, we refer to [15].
The differential ∂ J induces homomorphisms between the cohomology groups As ∂ 2 J = 0, we may ask whether the above sequence has any chance to be exact. Even if this fails, we can still try to quantify the deviation from exactness. For this purpose, we define the complex vector spaces: for all p. The groups E p,0 2 measure the deviation from exactness of the above sequence. As in the present work, we just need the groups E p,0 1 and E p,0 2 , we may define the groups E p,0 r+1 as the cohomology groups of the sequence As in the complex setup, we call {E p,0 r } p∈N0 the r th page of the quaternionic Frölicher spectral sequence. The quaternionic Frölicher spectral sequence is said to degenerate at the r th page if E p,0 r ≃ E p,0 r+1 for all p (note that then also E p,0 r ≃ E p,0 r+i for all p and all i 0). The next Lemma gives a sufficient condition for the quaternionic Frölicher spectral sequence to degenerate at the first page: Proof. Let a ∈ E p,0 1 be represented by ϕ. Then ∂ J ϕ lies in B p+1,0 . By hypothesis A p+1,0 ≃ B p+1,0 , so it also sits inside A p+1,0 and hence is ∂-exact.
However, this condition is not necessary as can be seen from Example 1 in Section 11 where we have a 3,0 = b 3,0 but nevertheless E 2,0 1 ≃ E 2,0 2 . In [17] a quaternionic version of the dd c -Lemma has been introduced: Definition 5. Let (M, I, J, K) be a hypercomplex manifold of real dimension 4n. We say that the ∂∂ J -Lemma holds if every ∂-closed, ∂ J -exact (p, 0)-form in Λ p,0 I (M ) is ∂∂ J -exact, for any 0 p 2n. This is equivalent to saying that every ∂ J -closed, ∂-exact form is ∂∂ J -exact.
Note that, by definition, the ∂∂ J -Lemma holds for (p, 0)-forms if and only if the vector space B p,0 (or equivalently D p,0 ) is trivial. In order to relate the ∂∂ J -Lemma to the quaternionic Frölicher spectral sequence, we need the following result: Proof. Let a ∈ H p,0 ∂ . We need to find a representative ϕ such that ∂ J ϕ = 0. The assumption E p,0 This leads to the following relation between the ∂∂ J -Lemma and the quaternionic Frölicher spectral sequence: Proof. If B p+1,0 ≃ 0, then A p+1,0 ≃ 0 by exactness of Varouchas' sequence. Lemma 4 then implies that E p,0 1 ≃ E p,0 2 . Conversely, we may apply Lemma 6 to deduce that E p,0 ≃ 0. The isomorphism ∂ J : E p,0 → B p+1,0 from Proposition 3 finishes the proof.
In words, if the ∂∂ J -Lemma holds, then the quaternionic Frölicher spectral sequence degenerates at the first page. However, in order to get the implication in the opposite direction, we need the additional assumption A p+1,0 ≃ 0.
Remark 8. Example 1 in Section 11 provides an example such that the quaternionic Frölicher spectral sequence degenerates at the first page but such that the ∂∂ J -Lemma does not hold.
We will return to the quaternionic Frölicher spectral sequence in Section 8 when we show that the quaternionic Frölicher spectral sequence always degenerates at the first page in real dimension 8 on manifolds with special holonomy, called SL(2, H)manifolds.

Quaternionic Frölicher-type inequality
In this Section, we give a quaternionic version of the Frölicher-type inequality proven by D. Angella and A. Tomassini in [5] in a very general setting (see also [4]). As in [5], we deduce a cohomological criterion for the quaternionic dd c -Lemma.
Since Λ p,0 I (M ), ∂, ∂ J is not a double complex, part of our proof differs slightly. For the reader's convenience, we give the details.
for any 0 p 2n.
Proof. By exactness of Varouchas' short exact sequences, we have 2 . The second equality follows from Proposition 3.
The following Theorem shows that equality happens precisely when the quaternionic dd c -Lemma is satisfied.
holds for all 0 p 2n.
Proof. If we assume the equality to hold, then the proof of Theorem 9 shows that both E p,0 for all p and a p,0 = 0 = f p,0 for all p. Theorem 7 implies that B p,0 ≃ 0 for all p and hence the ∂∂ J -Lemma holds. Conversely, if the ∂∂ J -Lemma holds, then b p,0 = 0 = d p,0 for all p. Exactness of Varouchas' sequences implies that a p,0 = 0 = f p,0 and by Lemma 4 we also get E p,0 1 ≃ E p,0 2 .

Metric structures
Up to this stage, the results have been cohomological in nature. We will now start to endow the hypercomplex manifolds with additional geometric structures. Recall the following special Hermitian metrics defined on complex manifolds, see for example [28]: In this short Section, we introduce the quaternionic counterparts of these metrics. A quaternionic Hermitian metric on a hypercomplex manifold (M, I, J, K) is a Riemannian metric g satisfying g(·, ·) = g(I·, I·) = g(J·, J·) = g(K·, K·).
The form is of Hodge type (2, 0) with respect to the complex structure I, i.e. Ω ∈ Λ 2,0 I (M ). Moreover, it is non-degenerate. Actually we can reconstruct g from Ω. Indeed, for any vector fields X, Y of type T 1,0 Definition 11. Let (M, I, J, K, Ω) be a quaternionic Hermitian manifold of real dimension 4n.
An important feature in non-Kähler complex geometry is that a complex manifold always admits a Hermitian Gauduchon metric. In the next Section we will see that, if we impose a constraint on the holonomy, then we similarly get existence of a quaternionic Gauduchon metric on hypercomplex manifolds.

SL(n, H)-manifolds
First, we recall that a hypercomplex manifold admits a canonical connection.

Theorem 12. [27]
A hypercomplex manifold (M, I, J, K) admits a unique torsionfree connection preserving all three complex structures I, J, K. This connection is called the Obata connection.
Since the Obata connection preserves the hypercomplex structure, its holonomy lies in the general quaternionic linear group GL(n, H) (see for instance [33]). However, in many examples such as nilmanifolds [7], the holonomy is actually contained in SL(n, H), the commutator subgroup of GL(n, H) (for more details see [39,37]). For any SL(n, H)-manifold, the canonical bundle Ω 2n,0 I (M ) is holomorphically trivial [39]. The converse is true if, moreover, the manifold is compact and admits an HKT metric [39]. We denote an SL(n, H)-manifold by (M, I, J, K, Φ) where Φ is a nowhere degenerate holomorphic section of Ω 2n,0 I (M ). Furthermore, we can assume that Φ satisfies Φ = JΦ. In particular, ∂Φ = ∂ J Φ = 0.
Whilst HKT metrics or quaternionic strongly Gauduchon metrics do not always exist on hypercomplex manifolds (see [14,34] and Remark 34) the following result shows that SL(n, H)-manifolds always admit a quaternionic Gauduchon metric. This crucial result is a first reason why in the sequel we will always assume to be on an SL(n, H)-manifold. The second reason is that it is possible to build a version of Hodge theory on SL(n, H)-manifolds [37] as shown next.

Pure and full hypercomplex structures
Besides the decomposition in self-dual and anti-self-dual forms, it is also possible to decompose (2, 0)-forms in real and imaginary forms. Indeed, any ϕ ∈ Λ where Ω J,+ 0 (M ) is the bundle of real forms pointwise h-orthogonal to Ω.
We saw that, on compact SL(2, H)-manifolds, the decomposition of (2, 0)-forms in self-dual and anti-self-dual forms passes to cohomology. In the remaining part of this Section, we will show that, on compact SL(2, H)-manifolds, the decomposition in real and imaginary forms descends to a splitting in cohomology as well. Remark 19. There are higher-dimensional SL(n, H)-manifolds with n > 2 whose hypercomplex structures are not C ∞ -pure-and-full: see Example 3 in Section 11.

Frölicher degeneracy on SL(2, H) manifolds
In this Section, we show that the quaternionic Frölicher spectral sequence always degenerates at the first page on compact SL(2, H)-manifolds: We prove now Theorem 20: Proof. If we show that the maps However, any ∂ J -exact * Φ -self-dual or * Φ -anti-self-dual (2, 0)-form is zero: For the third map, take a ∈ H 3,0 ∂ represented by ϕ such that ϕ = ∂ J α for some α representing a class b in H 2,0 ∂ . By Theorem 18, the hypercomplex structure of a compact SL(2, H)-manifold is C ∞ -pure-and-full. We deduce that either b ∈ H J,+

"non-HKT-ness" degrees
Similarly to the non-Kähler-ness degrees introduced in [3] by Angella-Dloussky-Tomassini on compact complex manifolds (see also [23,35]), we define non-HKTness degrees ∆ p on compact hypercomplex manifolds of real dimension 4n: where the last equality comes from SL(2, H)-symmetry. Moreover, on a compact hypercomplex manifold the sequence Using Theorem 9 and Theorem 27 we get The first part ∆ 2 ∈ {0, 2} follows because ∆ 2 is even. For the second part, recall that Theorem 23 assures that compact SL(2, H) HKT implies ∆ 2 = 0. Conversely, if we assume that ∆ 2 = 0, then h 2,0 BC = h 2,0 ∂ . Recall that Theorem 20 implies E 2,0 1 ≃ E 2,0 2 and that we get from Theorem 29 that a 3,0 = 0. We deduce from Theorem 7 that b 3,0 = 0 and from Proposition 3 that e 2,0 = 0. Using Varouchas' exact sequences we conclude that the map H 2,0 BC → H 2,0 ∂ is surjective and thus also injective. We then deduce from the above exact sequence that h 1,0 AE = h 1,0 ∂ . As J defines a quaternionic structure on H 1,0 AE , the dimension h 1,0 AE is even and we conclude from Theorem 26 that M is HKT.

Quaternionic strongly Gauduchon metrics
By definition, any HKT metric is a quaternionic strongly Gauduchon metric. In this Section we show that, on compact SL(2, H)-manifolds, both notions are actually equivalent: To imitate D. Popovici's proof in the complex realm, we will need the following result whose proof follows from Lemma 28.
Lemma 33. On compact hypercomplex manifolds, the sequence We have now all the tools to prove Theorem 31: Proof. Suppose that Ω is a strongly Gauduchon metric such that ∂ J Ω = ∂β for some β ∈ Λ 2,0 I (M ). We will prove that the map H 1,0 ∂ ∂J −→ H 2,0 BC in the exact sequence of Lemma 33 vanishes. Let α be a representative of an element a of H 1,0 ∂ : Using elliptic theory (see [17,Theorem 4.6]), there exists a real smooth function f such that (∂ J α + ∂∂ J f )∧Ω∧Φ = 0. This implies that we can choose a representative of a such that ∂ J α ∧ Ω ∧ Φ = 0, or equivalently such that ∂ J α ∧ Ω = 0. By a quaternionic version of the Hodge-Riemann relations [40]: Remark 34. There are compact hypercomplex manifolds which do not admit a quaternionic strongly Gauduchon metric, see Example 1 in Section 11.

Examples
In this Section we will compute the quaternionic Bott-Chern and Aeppli cohomologies for some left-invariant hypercomplex structures on nilmanifolds.
Let M = Γ\G be a nilmanifold with G a connected nilpotent Lie group equipped with a left-invariant hypercomplex structure (I, J, K) and Γ a discrete and cocompact subgroup of G. As the hypercomplex structure descends to the quotient M = Γ\G, the manifold M is called a hypercomplex nilmanifold. It has been proven in [7] that the holonomy of the Obata connection of a hypercomplex nilmanifold of real dimension 4n lies in SL(n, H). A way to construct such hypercomplex nilmanifolds is to consider a nilpotent Lie algebra g equipped with a left-invariant hypercomplex structure (I, J, K) and rational structure constants. Then, by a result of Mal'cev [24], the corresponding simply-connected nilpotent Lie group G admits a lattice Γ.
We briefly recall some known facts about Dolbeault cohomologies of complex nilmanifolds. Let (M, I) be a complex nilmanifold with left-invariant complex structure I. Then it is conjectured that the Dolbeault cohomology is isomorphic to the Dolbeault cohomology H p,q ∂ (g C ) of G-left-invariant forms, where g C is the complexification of g the Lie algebra of G. This isomorphism has been proven in several cases and we refer the reader to [11,12,29,30,31]. In particular, S. Console and A. Fino [11] have proven the isomorphism when the complex structure is rational, i.e. when J (g Q ) ⊆ g Q , where g Q is the rational structure induced by Γ (see [11]).
Under the same conditions, we also have the isomorphism Proof. Recall that complex conjugation gives an isomorphism between H p,0 ∂ (M ) (resp. H p,0 ∂ (g C )) and H 0,p ∂ (M ) (resp. H 0,p ∂ (g C )). Moreover, Proposition 3 gives an isomorphism between H p,0 ∂ (M ) (resp. H p,0 ∂ (g C )) and H p,0 ∂J (M ) (resp. H p,0 ∂J (g C )) induced by the map J. To get the isomorphism for the quaternionic Frölicher spectral sequence, note that the map of complexes i : (Λ p,0 . An isomorphism for the first page then leads to an isomorphism for all subsequent pages, see [25,Theorem 3.4]. In the complex setup, D. Angella proved that, if the De Rham and Dolbeault cohomologies of the complex nilmanifold (M, I) are isomorphic to the corresponding cohomologies of G-left-invariant forms, then the Bott-Chern and Aeppli cohomologies of (M, I) can also be computed using only G-left-invariant forms [2]. In the quaternionic setup, the proof of injectivity from [2, Lemma 3.6] (see also [11]) can readily be adapted. We deduce that the quaternionic Bott-Chern cohomology H p,0 BC (g C ) can be injected in H p,0 BC (M ) and similarly for the Aeppli cohomology. The next result proves surjectivity in the special case when the hypercomplex nilmanifold has real dimension 8.
Proof. To prove the isomorphism for Bott-Chern cohomology, we use Varouchas' , thus it is sufficient to prove that D p,0 (M ) can be computed using only G-left-invariant forms. It follows from Lemma 28 and Proposition 3 that D 1,0 (M ), and hence also H 1,0 BC (M ), can be computed using only G-left-invariant forms (this is actually true in any real dimension 4n). Since M is a compact SL(2, H)-manifold, Theorem 29 implies f 1,0 = f 3,0 = a 3,0 = 0. We deduce from Varouchas' sequences (Lemma 2) and Corollary 35 that the spaces E 1,0 (M ), E 3,0 (M ) and B 3,0 (M ) can be computed using only G-left-invariant forms. Therefore, using Proposition 3, we conclude the same for D 2,0 (M ), D 3,0 (M ) and D 4,0 (M ). Finally, we point out that D 0,0 (M ) is trivial.
For the isomorphism of Aeppli cohomology, note that the Hodge star operator associated to an invariant metric * Φ : Λ p,0 AE (g C ) in cohomology. As we have already proven the isomorphism for the quaternionic Bott-Chern cohomology, the commutative diagram 11.1. Example 1. The first Example we consider is a hypercomplex nilmanifold of real dimension 8. It is built out of a central extension of the quaternionic Heisenberg algebra R × H 7 and already appeared in [14]. We will see that it is an SL(2, H)manifold which does not admit any HKT metric. The structure equations of the Lie algebra in question are: We consider the following hypercomplex structure: If we express the structure equations in terms of the (1, 0)-forms we get: where we use the notations ϕ 12 = ϕ 1 ∧ ϕ 2 and ϕ 1 = (ϕ 1 ) = e 1 + ie 2 . We see that dϕ 4 ⊆ Λ 1,1 I and hence the hypercomplex structure is not abelian [14]. If we rewrite the structure equations in terms of the differentials ∂ and ∂ J we get: This leads to the following tables: We point out the following observations. Firstly, one checks that, in accordance with Theorem 20, the quaternionic Frölicher spectral sequence degenerates at the first page. Secondly, in accordance with Theorem 18, the hypercomplex structure is C ∞ -pure-and-full. Indeed, the space H 2,0 ∂ = ϕ 12 ,ϕ 13 ,ϕ 14 ,ϕ 23 ,ϕ 24 Thirdly, as predicted by Theorem 29, we have ∆ 1 = ∆ 3 = 0. Lastly, the second non-HKT-ness degree is non-zero ∆ 2 = 2 and hence, by Theorem 30, this hypercomplex nilmanifold does not admit any HKT metric.
11.2. Example 2. The second Example is also based upon the central extension of the quaternionic Lie algebra R × H 7 but now we consider a path of hypercomplex structures as done in [14] and [17]. We end up with an SL(2, H)-manifold carrying a family t ∈ (0, 1) of hypercomplex structures which is HKT for t = 1 2 but not HKT for all other values of t. The structure equations of the Lie algebra are the same as before:        de 1 = de 2 = de 3 = de 4 = de 5 = 0 de 6 = e 1 ∧ e 2 + e 3 ∧ e 4 de 7 = e 1 ∧ e 3 + e 4 ∧ e 2 de 8 = e 1 ∧ e 4 + e 2 ∧ e 3 We consider the family of hypercomplex structures (I t , J t , K t ) parametrised by t ∈ (0, 1): I t e 1 = t−1 t e 2 I t e 3 = e 4 I t e 5 = 1 t e 6 I t e 7 = e 8 J t e 1 = t−1 t e 3 J t e 2 = −e 4 J t e 5 = 1 t e 7 J t e 6 = −e 8 A basis of left-invariant (1, 0)-forms is given by: If t = 1 2 , then dϕ i ⊆ Λ 1,1 I and the complex structure is abelian whereas otherwise it is not. In terms of the differentials ∂ and ∂ J , the structure equations can be written as: We conclude: if t = 1 2 then we get the same tables as in Example 1. On the other hand, if t = 1 2 then both ∂ϕ 4 = 0 and ∂ J ϕ 3 = 0. Consequently the spaces A p,0 , B p,0 , C p,0 , D p,0 , E p,0 and F p,0 are trivial while We deduce that, just as the HKT property, the ∂∂ J -Lemma is not stable:
This differs from the complex case where the ∂∂-Lemma is stable by small complex deformations [4,41,42]. Note that we can also use this Example to show that, even on compact SL(2, H)-manifolds, the second non-HKT-ness degree is not stable by small hypercomplex deformations, whereas on compact complex surfaces, the second non-Kähler-ness degree is a topological invariant, see [6, Corollary 1.2]. 11.3. Example 3. The third Example we consider is an SL(3, H)-manifold whose hypercomplex structure is neither C ∞ -pure nor C ∞ -full. The structure equations of the nilpotent Lie algebra in question are