About Zero Torsion Linear Maps on Lie Algebras

Abstract

We prove that any zero torsion linear map on a nonsolvable real Lie algebra 𝔤₀ is an extension of some CR-structure. We then study the cases of 𝔰𝔩(2, ℝ) and the 3-dimensional Heisenberg Lie algebra 𝔫. In both cases, we compute up to equivalence all zero torsion linear maps on 𝔤₀, and deduce an explicit description of the equivalence classes of integrable complex structures on 𝔤₀×𝔤₀.

1. Introduction

Given a real Lie algebra 𝔤₀, the determination up to equivalence of zero torsion linear maps from 𝔤₀ to 𝔤₀ plays an important role in the computation of complex structures on direct products involving 𝔤₀ [1]. The direct computation of those maps can be difficult for semisimple 𝔤₀, so there is a point in exploring alternative ways, particularly their relation to 𝐶𝑅-structures. For compact 𝔤₀, maximal rank 𝐶𝑅-structures have been classified up to equivalence in [2]. In the case of 𝔰𝔲(2), all zero torsion linear maps are extensions of certain 𝐶𝑅-structures (see [1]). One can then ask the natural question whether or not any zero torsion linear map on a nonabelian 𝔤₀ is necessarily an extension of some 𝐶𝑅-structure. In the present note, we answer the question in the positive for nonsolvable Lie algebras. Then we make a detailed study of two basic examples: 𝔤₀=𝔰𝔩(2,ℝ) in the positive case, and 𝔤₀=𝔫 the 3-dimensional Heisenberg Lie algebra in the negative. In both cases, we compute (up to equivalence) all zero torsion linear maps, and the result is used to exhibit a complete set of representatives of equivalence classes of complex structures on 𝔤₀×𝔤₀.

An interesting direction for future research could be to investigate zero torsion linear maps and 𝐶𝑅-structures on various constructions of Lie algebras, for example like those considered in [3] (see also [4]).

2. Zero Torsion Linear Maps and Extension of 𝐶𝑅-Structures

A 𝐶𝑅-structure on a smooth real manifold 𝑀 is a subbundle 𝒱 of the complexified tangent bundle ℂ𝑇(𝑀) of 𝑀 such that [𝒱,𝒱]⊂𝒱 (i.e., the space of smooth sections of 𝒱 is closed under commutators) and 𝒱∩𝒱={0} ( ⁻ denoting here conjugation in ℂ𝑇(𝑀)). The rank or 𝐶𝑅-dimension is the complex dimension of 𝒱. For general background on 𝐶𝑅-structures we refer the reader to [5].

Throughout this section, 𝔤₀ will denote any finite-dimensional real Lie algebra, 𝔤 its complexification, and 𝜎 or simply ⁻ the conjugation in 𝔤 with respect to 𝔤₀.

If 𝐺₀ is a connected finite dimensional real Lie group, with Lie algebra 𝔤₀, left invariant 𝐶𝑅-structures on 𝐺₀ are identified to 𝐶𝑅-structures on 𝔤₀ in the following sense [6, 7].

Definition 2.1. A rank 𝑟(1⩽𝑟⩽[(dim𝔤₀)/2])𝐶𝑅-structure on 𝔤₀ is a 𝑟-dimensional complex subalgebra 𝔪 of 𝔤 such that 𝔪∩𝔪={0}.

If dim𝔤₀ is even, a 𝐶𝑅-structure of maximal rank is an (integrable) complex structure.

Now one has the following straightforward lemma.

Lemma 2.2. Let 𝔭 be vector subspace of 𝔤₀ and 𝐽_𝔭∶𝔭→𝔭 a linear map. Consider the real vector subspace 𝔪={𝑋;𝑋∈𝔭} of 𝔤, where 𝑋=𝑋−𝑖𝐽_𝔭𝑋 for 𝑋∈𝔭. Denote 𝔪^=𝑖𝔪={𝐽_𝔭𝑋+𝑖𝑋;𝑋∈𝔭}. Then (i)𝔪 is a complex vector subspace of 𝔤 if and only if 𝐽^2𝔭=−1_𝔭;(ii)𝔪 is stable with respect to the bracket if and only if for any𝑋,𝑌∈𝔭𝐽[𝑋,𝑌]−𝐽_𝔭𝑋,𝐽_𝔭𝑌∈𝔭,_𝔭[𝑋,𝑌]−𝐽_𝔭𝑋,𝐽_𝔭𝑌=𝐽_𝔭𝑋,𝑌+𝑋,𝐽_𝔭𝑌.(2.1) In that case, [𝑋,𝑌]=𝑍 with 𝑍=[𝑋,𝑌]−[𝐽_𝔭𝑋,𝐽_𝔭𝑌].(iii)𝔪^ is stable with respect to the bracket if and only if for any 𝑋,𝑌∈𝔭𝐽𝐽_𝔭𝑋,𝑌+𝑋,𝐽_𝔭𝑌∈𝔭,_𝔭𝐽_𝔭𝑋,𝑌+𝑋,𝐽_𝔭𝑌=𝐽_𝔭𝑋,𝐽_𝔭𝑌−[𝑋,𝑌].(2.2) In that case, [𝑋,𝑌]=−𝑖𝑍 with 𝑍=[𝐽_𝔭𝑋,𝑌]+[𝑋,𝐽_𝔭𝑌] for 𝑋,𝑌∈𝔭.

Remark 2.3. When (i) is satisfied, (ii) and (iii) are trivially equivalent since then 𝔪=𝔪^.

Lemma 2.4. A rank 𝑟𝐶𝑅-structure on 𝔤₀ can be defined in an alternative way as (𝔭,𝐽_𝔭) where 𝔭 is a 2𝑟-dimensional (1⩽𝑟⩽[(dim𝔤₀)/2]) vector subspace of 𝔤₀ and 𝐽_𝔭∶𝔭→𝔭 is a linear map satisfying the 3 conditions 𝐽𝐽^2𝔭=−1_𝔭,(2.3)_𝔭𝐽𝑋,𝑌+𝑋,𝐽_𝔭𝑌∈𝔭∀𝑋,𝑌∈𝔭,(2.4)_𝔭𝑋,𝐽_𝔭𝑌−[𝑋,𝑌]−𝐽_𝔭𝐽_𝔭𝑋,𝑌+𝑋,𝐽_𝔭𝑌=0∀𝑋,𝑌∈𝔭.(2.5)

Proof. Let 𝔪 be a rank 𝑟𝐶𝑅-structure on 𝔤₀. Note first that the taking of the real part is a bijective linear map of the real algebra 𝔪 onto its image 𝔭=ℜ𝔪,  dim𝔭=2𝑟, and there exists a unique linear map 𝐽_𝔭∶𝔭→𝔤₀ such that 𝔪={𝑋=𝑋−𝑖𝐽_𝔭𝑋;𝑋∈𝔭}. Now, for 𝑋∈𝔭,  𝑖𝑋=𝐽_𝔭𝑋+𝑖𝑋∈𝑖𝔪=𝔪 hence 𝐽_𝔭𝑋∈𝔭, so that 𝐽_𝔭∶𝔭→𝔭. Then (2.3), (2.4), (2.5) follow from Lemma 2.2 and Remark 2.3.
The converse comes again from Lemma 2.2 and Remark 2.3.

Remark 2.5. The condition (2.3) implies det𝐽_𝔭=1 and Trace (𝐽_𝔭)=0, hence if 𝑟=1, (2.4) and (2.5) follow from (2.3) and can be omitted.

Definition 2.6. A linear map 𝐽∶𝔤₀→𝔤₀ is said to have zero torsion if it satisfies the condition [𝐽𝑋,𝐽𝑌]−[𝑋,𝑌]−𝐽[𝐽𝑋,𝑌]−𝐽[𝑋,𝐽𝑌]=0∀𝑋,𝑌∈𝔤₀.(2.6)

If 𝐽 has zero torsion and satisfies in addition 𝐽²=−1,  𝐽 is an (integrable) complex structure on 𝔤₀. That means that 𝐺₀ can be given the structure of a complex manifold with the same underlying real structure and such that the canonical complex structure on 𝐺₀ is the left invariant almost complex structure 𝐽 associated to 𝐽 (for more details, see [8]).

When computing the matrices of the zero torsion maps in some fixed basis (𝑥_𝑗)_{1⩽𝑗⩽𝑛} of 𝔤₀, we will denote by 𝑖𝑗∣𝑘 (1⩽𝑖,𝑗,𝑘⩽𝑛) the torsion equation obtained by projecting on 𝑥_𝑘 the equation (2.6) with 𝑋=𝑥_𝑖, 𝑌=𝑥_𝑗.

The automorphism group Aut  𝔤₀ of 𝔤₀ acts on the set of all zero torsion linear maps and on the set of all complex structures on 𝔤₀ by 𝐽↦Φ∘𝐽∘Φ⁻¹ for all Φ∈Aut  𝔤₀. It acts also on the set of 𝐶𝑅-structures by (𝔭,𝐽_𝔭)↦(Φ𝔭,Φ∘𝐽_𝔭∘Φ⁻¹). Two 𝐽,𝐽^ (resp., (𝔭,𝐽_𝔭), (𝔭^,𝐽_𝔭′)) are said to be equivalent (notation: 𝐽≡𝐽^ (resp., 𝐽_𝔭≡𝐽_𝔭′)) if they are on the same Aut  𝔤₀ orbit. This means that the corresponding left invariant 𝐶𝑅-structures on the connected simply connected real Lie group associated to 𝔤₀ are intertwined by some Lie group automorphism. It is a stronger notion than 𝐶𝑅-diffeomorphy, where the intertwining is simply required to be a diffeomorphism.

Lemma 2.7. Let 𝐽∶𝔤₀→𝔤₀ be a linear map, 𝔪={𝑋=𝑋−𝑖𝐽𝑋;𝑋∈𝔤₀} and 𝔪^=𝑖𝔪={𝐽𝑋+𝑖𝑋;𝑋∈𝔤₀}; (i)𝔪∩𝔪^={𝑋;𝑋∈ker(𝐽²+1)}, (ii)𝔪^ is a real subalgebra of 𝔤 if and only if 𝐽 has zero torsion,(iii)if 𝐽 has zero torsion, 𝔪∩𝔪^ is a complex subalgebra of 𝔤.

Proof. (i) For any 𝑍∈𝔤 one has𝑍∈𝔪∩𝔪^⟺⟺∃∃𝑋𝑋,,𝑌𝑌∈∈𝔤𝔤₀₀,𝑍=𝑋−𝑖𝐽𝑋=𝐽𝑌+𝑖𝑌⟺∃𝑋∈𝔤₀,𝑍=𝑋−𝑖𝐽𝑋,𝑋=𝐽Y,𝑌=−𝐽𝑋⟺∃𝑋∈ke,r𝑍𝐽₂=𝑋−𝑖𝐽𝑋,𝑋=−𝐽²𝑋+1,𝑍=𝑋.(2.7)
(ii) The result follows from Lemma 2.2(iii) since the first condition in (2.2) (with𝔭=𝔤₀ and 𝐽_𝔭=𝐽) is trivially satisfied and the second condition is the zero torsion condition.
(iii) From (ii), [𝔪^,𝔪^]⊂𝔪^, hence [𝔪,𝔪^]⊂𝔪 and [𝔪∩𝔪^,𝔪∩𝔪^]⊂𝔪∩𝔪^. Clearly 𝔪∩𝔪^ is stable by multiplication by 𝑖.

Definition 2.8. Let 𝐽∶𝔤₀→𝔤₀ be a zero torsion linear map. We say that 𝐽 is an extension of a 𝐶𝑅-structure if there exists a vector subspace 𝔭≠{0} of 𝔤₀ such that 𝔭 equipped with the restriction 𝐽_𝔭 of 𝐽 to 𝔭 is a 𝐶𝑅-structure on 𝔤₀.

Definition 2.9. A real form 𝔲 of 𝔤 is said to be of type I (with respect to 𝔤₀) if 𝔤₀∩𝔲≠{0}.  𝔤₀ is said to be type I if any real form 𝔲 of 𝔤 is of type I.

Remark 2.10. Introduce the real linear projections 𝜋₁∶𝔤→𝔤₀, 𝜋₂∶𝔤→𝔤₀ defined by 𝑧=𝜋₁(𝑧)+𝑖𝜋₂(𝑧) for 𝑧∈𝔤. Then a real form 𝔲 of 𝔤 is of type I if and only if ker𝜋₂∣_𝔲≠{0}.

Proposition 2.11. Let 𝐽∶𝔤₀→𝔤₀ be a zero torsion linear map, 𝔪={𝑋=𝑋−𝑖𝐽𝑋;𝑋∈𝔤₀} and 𝔪^=𝑖𝔪.  𝐽 is an extension of a 𝐶𝑅-structure if and only if 𝔪∩𝔪^≠{0}.

Proof. From Lemma 2.7, 𝔪∩𝔪^ is a complex subalgebra of 𝔤 and 𝔪∩𝔪^={𝑋;𝑋∈ker(𝐽²+1)}. If 𝐽 is an extension of a 𝐶𝑅-structure, one has {0}≠𝔭⊂ker(𝐽²+1) hence 𝔪∩𝔪^≠{0}. Conversely, if 𝔪∩𝔪^≠{0}, let 𝔭=ker(𝐽²+1). Then 𝔭 is stable under 𝐽, and if 𝐽_𝔭 denotes the restriction of 𝐽 to 𝔭, conditions (2.3), (2.5) are trivially satisfied. Condition (2.4) holds true since, from Lemma 2.7(iii), 𝔪∩𝔪^ is a subalgebra of 𝔤. Precisely, for 𝑋,𝑌∈𝔭,  𝑋,𝑌∈𝔪∩𝔪^ hence [𝑖𝑋,𝑌]=[𝐽_𝔭𝑋+𝑖𝑋,𝑌−𝑖𝐽_𝔭𝑌]∈𝔪∩𝔪^ and (2.4) follows.

Corollary 2.12. There is a one-to-one correspondence between nontype I real forms of 𝔤 and zero torsion linear maps 𝐽∶𝔤₀→𝔤₀ which are no extension of a 𝐶𝑅-structure.

Proof. Let 𝐽∶𝔤₀→𝔤₀ be a zero torsion linear map that is no extension of a 𝐶𝑅-structure. Then 𝔪∩𝔪^={0}, hence 𝔪^={𝐽𝑋+𝑖𝑋;𝑋∈𝔤₀} is a real form of 𝔤 which is nontype I. Conversely, if 𝔲 is a nontype I real form of 𝔤, then 𝜋₂(𝔲)=𝔤₀ implies 𝔲={𝐽𝑋+𝑖𝑋;𝑋∈𝔤₀} for some linear map 𝐽∶𝔤₀→𝔤₀, that is, 𝔲=𝔪^. As 𝔪^ is a real subalgebra, 𝐽 has zero torsion from Lemma 2.7(ii). Now 𝔪∩𝔪^={0} since 𝔲 is a real form hence 𝐽 is not an extension of a 𝐶𝑅-structure.

Corollary 2.13. If 𝔤₀ is of type I, then any zero torsion linear map 𝐽∶𝔤₀→𝔤₀ is an extension of a 𝐶𝑅-structure.

Proposition 2.14. Let 𝔲 be any real form of 𝔤,  𝜏,  𝜎 the conjugations with respect to 𝔲,  𝔤₀, and 𝑁=𝜎𝜏∈Aut𝔤. If 𝑁 has a nonzero fixed point, then 𝔲 is type I.

Proof. Let 𝑍 be a fixed point of 𝑁. 𝑁𝑍=𝑍 reads 𝜎𝑍=𝜏𝑍. Consider 𝑉=𝜎𝑍=𝜏𝑍. Then 𝜎𝑉=𝜏𝑉=𝑍. Hence 𝑊=𝑉+𝑍 has 𝜏𝑊=𝑍+𝜏𝑍=𝑍+𝑉=𝑊 and similarly 𝜎𝑊=𝑊. Hence 𝑊∈𝔤₀∩𝔲. Now, 𝑊=0 if and only if 𝜎𝑍=𝜏𝑍=−𝑍, that is, 𝑖𝑍∈𝔤₀∩𝔲.

Corollary 2.15. If 𝔤₀ is nonsolvable, then it is type I.

Proof. If 𝔤₀ is nonsolvable, so is 𝔤. Now, it is known that any automorphism of a nonsolvable Lie algebra over a characteristic 0 field has a nonzero fixed point ([9]). Hence any real form 𝔲 of 𝔤 is type I.

3. Case of 𝔰𝔩(2,ℝ)

𝐺=𝑆𝐿(2,ℝ) denotes the Lie group of real 2×2 matrices with determinant 1𝜎=𝑎𝑐𝑑𝑏,𝑎𝑑−𝑏𝑐=1.(3.1) Its Lie algebra 𝔤₀=𝔰𝔩(2,ℝ) consists of the zero trace real 2×2 matrices 𝑋=𝑥𝑧−𝑦𝑥=𝑥𝐻+𝑦𝑋₊+𝑧𝑋₋,(3.2) with basis 𝐻=¹⁰⁰⁻¹, 𝑋₊=⁰¹⁰⁰, 𝑋₋=⁰⁰¹⁰ and commutation relations𝐻,𝑋₊=2𝑋₊,𝐻,𝑋₋=−2𝑋₋,𝑋₊,𝑋₋=𝐻.(3.3) Beside the basis (𝐻, 𝑋₊, 𝑋₋), we will also make use of the basis (𝑌₁,𝑌₂,𝑌₃) where 𝑌₁=(1/2)𝐻,  𝑌₂=(1/2)(𝑋₊−𝑋₋),  𝑌₃=(1/2)(𝑋₊+𝑋₋), with commutation relations𝑌₁,𝑌₂=𝑌₃,𝑌₁,𝑌₃=𝑌₂,𝑌₂,𝑌₃=𝑌₁.(3.4) The adjoint representation of 𝐺 on 𝔤₀ is given by Ad(𝜎)𝑋=𝜎𝑋𝜎⁻¹. The matrix Φ of Ad(𝜎) (𝜎 as in (3.1)) in the basis (𝐻,𝑋₊,𝑋₋) isΦ=⎛⎜⎜⎜⎝1−+22𝑎𝑏𝑏𝑐−𝑎𝑎²𝑐−𝑏𝑏𝑑²2𝑐𝑑−𝑐²𝑑²⎞⎟⎟⎟⎠.(3.5) The adjoint group Ad(𝐺) is the identity component of Aut  𝔤₀ and one hasAut𝔤₀=Ad(𝐺)∪Ψ₀Ad(𝐺),Ψ₀=diag(1,−1,−1).(3.6) The adjoint action of 𝐺 on 𝔤₀ preserves the form 𝑥²+𝑦𝑧. The orbits are as follows: (i)the trivial orbit {0};(ii)the upper sheet 𝑧>0 of the cone 𝑥²+𝑦𝑧=0 (orbit of 𝑋₋); (iii)the lower sheet 𝑧<0 of the cone 𝑥²+𝑦𝑧=0 (orbit of −𝑋₋); (iv)for all 𝑠>0 the one-sheet hyperboloid 𝑥²+𝑦𝑧=𝑠² (orbit of 𝑠𝐻); (v)for all 𝑠>0 the upper sheet 𝑧>0 of the hyperboloid 𝑥²+𝑦𝑧=−𝑠² (orbit of 𝑠(−𝑋₊+𝑋₋)); (vi)for all 𝑠>0 the lower sheet 𝑧<0 of the hyperboloid 𝑥²+𝑦𝑧=−𝑠² (orbit of 𝑠(𝑋₊−𝑋₋)).

The orbits of 𝔤₀ under the whole Aut𝔤₀ are as follows, beside {0}: (I)the cone 𝑥²+𝑦𝑧=0 (orbit of 𝑋₋); (II)the one-sheet hyperboloid 𝑥²+𝑦𝑧=𝑠² (orbit of 𝑠H) (𝑠>0); (III) the two-sheet hyperboloid 𝑥²+𝑦𝑧=−𝑠² (orbit of 𝑠(𝑋₊−𝑋₋)) (𝑠>0).

Proposition 3.1. Let 𝔤₀=𝔰𝔩(2,ℝ), and 𝐽∶𝔤₀→𝔤₀ any linear map. 𝐽 has zero torsion if and only if it is equivalent to the endomorphism defined in the basis (𝑌₁,𝑌₂,𝑌₃)(resp., (𝐻,𝑋₊,𝑋₋)) by 𝐽_∗(𝜆)=⎛⎜⎜⎜⎝001𝜆00−001⎞⎟⎟⎟⎠,𝜆∈ℝ,(3.7)𝐽_∗(𝜆)≢𝐽_∗(𝜇) for 𝜆≠𝜇 (resp., 𝐽(𝛼)=⎛⎜⎜⎜⎝0−111−𝛼𝛼2−−𝛼𝛼12⎞⎟⎟⎟⎠,𝛼∈ℝ,(3.8)𝐽(𝛼)≢𝐽(𝛽) for 𝛼≠𝛽).

Proof. Let 𝐽=(𝜉^𝑖𝑗)_{1⩽𝑖,𝑗⩽3} in the basis (𝐻,𝑋₊,𝑋₋). The 9 torsion equations are in the basis (𝐻,𝑋₊,𝑋₋): 1122∣∣1222𝜉𝜉²²₂₁+𝜉¹¹𝜉¹²+𝜉²²−𝜉¹¹𝜉³¹−𝜉²¹+2𝜉¹³𝜉³²=0,12∣3𝜉₃₁𝜉¹²+1+𝜉²²²−𝜉³¹𝜉²¹−2𝜉³²𝜉²³=0,13∣1𝜉₂₁+2𝜉¹²𝜉³¹−2𝜉²²+2𝜉¹¹𝜉³²+2𝜉³³𝜉³²=0,13∣22𝜉₂₂−2𝜉¹³𝜉¹¹+2𝜉²³𝜉¹²+𝜉³¹𝜉²³−𝜉²¹+2𝜉¹³𝜉³³=0,13∣3𝜉₃₁−2𝜉¹¹𝜉²³+𝜉²¹+2𝜉¹³𝜉²¹−2𝜉³³𝜉²³=0,23∣14𝜉𝜉²¹₁₃−2𝜉³¹𝜉¹³−2+2𝜉³²𝜉²³−2𝜉³³²=0,23∣24𝜉₂₃𝜉¹²−1−𝜉²²𝜉¹¹−𝜉³²𝜉²³+𝜉²²−𝜉¹¹𝜉³³=0,23∣34𝜉₃₂𝜉¹²−𝜉²²+𝜉³³𝜉²¹=0,𝜉¹³−𝜉²²+𝜉³³𝜉³¹=0.(3.9)𝐽 has at least one real eigenvalue 𝜆. Let 𝑣∈𝔤₀,  𝑣≠0, an eigenvector associated to 𝜆. From the classification of the Aut𝔤₀ orbits of 𝔤₀, we then get 3 cases according to whether 𝑣 is on the orbit (I), (II), (III)  (in the cases (II), (III) one may choose 𝑣 so that 𝑠=1). Case 1. There exists 𝜑∈Aut𝔤₀ such that 𝑣=𝜑(𝑋₋). Then, replacing 𝐽 by 𝜑⁻¹𝐽𝜑, we may suppose 𝜉¹³=𝜉²³=0. That case is impossible from 13∣2 and 13∣3.Case 2. There exists 𝜑∈Aut𝔤₀ such that 𝑣=𝜑(𝐻). Then we may suppose 𝜉²¹=𝜉³¹=0. Then from 12∣2, 𝜉²³𝜉³²≠0, and 23∣2, 23∣3 yield 𝜉¹²=𝜉¹³=0. Then 12∣3 and 13∣2 successively give 𝜉³³=𝜉²²+2𝜉¹¹ and 𝜉¹¹=0. Now 12∣2 and 23∣1 read respectively −𝜉²³𝜉³²+(𝜉²²)²+1=0, and 𝜉²³𝜉³²−(𝜉²²)²+1=0. Hence that case is impossible.Case 3. There exists 𝜑∈Aut𝔤₀ such that 𝑣=𝜑(𝑋₊−𝑋₋). Then we may suppose that 𝑣=𝑋₊−𝑋₋. Now instead of the basis (𝐻,𝑋₊,𝑋₋), we consider the basis (𝑌₁,𝑌₂,𝑌₃). The matrix of 𝐽 in the basis (𝑌₁,𝑌₂,𝑌₃) has the form𝐽_∗=⎛⎜⎜⎜⎝𝜂¹¹0𝜂¹³𝜂²¹𝜆𝜂²³𝜂³¹0𝜂²³⎞⎟⎟⎟⎠.(3.10) Then the 9 torsion equations ∗𝑖𝑗∣𝑘 (the star is to underline that the new basis is in use) for 𝐽 in that basis are as follows: ∗∗1122∣∣12𝜂𝜂³¹₁₁+𝜂¹³𝜆−𝜂³¹−𝜂¹³𝜂¹¹=0,∗12∣3𝜂₁₁+𝜆𝜂²³−𝜂²¹𝜂³¹=0,∗13∣1𝜂₂₃𝜆−1+𝜂³¹²−𝜂¹¹+𝜆𝜂³³=0,∗13∣2𝜂₁₁𝜂¹³+𝜂²¹𝜂¹¹+𝜂²³𝜂³¹−𝜂²¹𝜂³³=0,∗13∣3𝜂₂₃𝜆+1+𝜂²¹²+𝜂²³²+𝜂³¹𝜂¹³−𝜂¹¹−𝜆𝜂³³=0,∗23∣1𝜂₁₁𝜂¹¹−𝜂²¹𝜂¹³+𝜂³¹−𝜂²³𝜂³³=0,∗23∣2𝜂₂₃𝜆+1−𝜂¹³²+𝜂¹¹−𝜆𝜂³³=0,∗23∣3𝜂𝜂¹³₃₁−𝜂³³+𝜆𝜂²¹=0,+𝜂¹³𝜆+𝜂³¹−𝜂¹³𝜂³³=0.(3.11) From ∗12∣1 and ∗23∣3, 𝜂¹¹𝜂³¹−𝜂¹³=−𝜂³³𝜂³¹−𝜂¹³.(3.12)Case 1. Suppose first that 𝜂³¹=𝜂¹³. Then 𝜆𝜂³¹=0.Subcase 1. Consider the subcase 𝜂³¹=0.∗13∣1 and ∗13∣3 read, respectively, (𝜂³³−𝜂¹¹)𝜂²¹=0,(𝜂³³−𝜂¹¹)𝜂²³=0. Suppose 𝜂³³≠𝜂¹¹. Then 𝜂²¹=𝜂²³=0, and ∗13∣2 gives 𝜂¹¹𝜆+1=(𝜂¹¹−𝜆)𝜂³³, which implies 𝜂³³=0 by ∗23∣1. As ∗12∣3 then reads 1=0, this case 𝜂³³≠𝜂¹¹ is not possible. Now, the case 𝜂³³=𝜂¹¹ is not possible either since then ∗23∣1 would read (𝜂¹¹)²+1=0. We conclude that the Subcase 1 is not possible. Subcase 2. Hence we are in the Subcase 2:𝜂³¹≠0. Then 𝜆=0. From ∗13∣2, 𝜂³³𝜂¹¹≠0. Then ∗23∣1 yields 𝜂³³=(−1+(𝜂³¹)²)/𝜂¹¹ and ∗13∣2 reads (𝜂²¹)²+(𝜂²³)²+2=0. This Subcase 2 is not possible either.Case 2. Hence Case 1 is not possible, and we are necessarily in the Case 2:𝜂³¹≠𝜂¹³. From (3.12), 𝜂³³=−𝜂¹¹. Then ∗13∣2 reads (𝜂¹¹)²+(𝜂²¹)²+(𝜂²³)²+1+𝜂³¹𝜂¹³=0 hence 𝜂³¹≠0 and 𝜂¹³=−((𝜂¹¹)²+(𝜂²¹)²+(𝜂²³)²+1)/𝜂³¹. From ∗12∣2,𝜂²¹=(𝜂²³(𝜂¹¹+𝜆))/𝜂³¹. Then ∗23∣2 reads 𝜂²³(((𝜂²³)²+𝜆²+1)(𝜂³¹)²+(𝜂¹¹+𝜆)²(𝜂²³)²)=0, that is, 𝜂²³=0, which implies 𝜂²¹=0. Now ∗12∣1 reads 𝜆(1+(𝜂¹¹)²−(𝜂³¹)²)=−𝜂¹¹(1+(𝜂¹¹)²+(𝜂³¹)²). The subcase 𝜂¹¹≠0 is not possible since then ∗12∣3 would yield 𝜆=−((𝜂¹¹)²+(𝜂³¹)²−1)/2𝜂¹¹ and ∗12∣1 would read ((𝜂¹¹)²+(𝜂³¹+1)²)((𝜂¹¹)²+(𝜂³¹−1)²)=0. Hence 𝜂¹¹=0. Then ∗12∣3 reads (𝜉³¹)²=1. The condition (𝜉³¹)²=1 now implies the vanishing of all the torsion equations. In that case 𝐽_∗=⎛⎜⎜⎜⎝00𝜀𝜆00−00𝜀⎞⎟⎟⎟⎠,𝜀=±1.(3.13) Then in the basis (𝐻,𝑋₊,𝑋₋)𝐽=⎛⎜⎜⎜⎜⎜⎝0−𝜀𝜀𝜆2−2𝜀𝜀−2𝜆−𝜆22𝜆2⎞⎟⎟⎟⎟⎟⎠.(3.14) The cases 𝜀=±1 are equivalent under Ψ₀.

Remark 3.2. 𝐽_∗(𝜆) is an extension of a 𝐶𝑅-structure, in agreement with Corollary 2.15.

4. Complex Structures on 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ)

We consider the basis (𝑌¹⁽¹⁾,𝑌²⁽¹⁾,𝑌³⁽¹⁾,𝑌¹⁽²⁾,𝑌²⁽²⁾,𝑌³⁽²⁾) of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ), with the upper index referring to the first or second factor. The automorphisms of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) fall into 2 kinds: the first kind is comprised of the diag(Φ₁,Φ₂), Φ₁,Φ₂∈Aut𝔰𝔩(2,ℝ), and the second kind is comprised of the Γ∘diag(Φ₁,Φ₂), with Γ the switch between the two factors of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ).

Proposition 4.1. Any integrable complex structure 𝐽 on 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) is equivalent under some first kind automorphism to the endomorphism given in the basis (𝑌¹⁽¹⁾,𝑌²⁽¹⁾,𝑌³⁽¹⁾,𝑌¹⁽²⁾,𝑌²⁽²⁾,𝑌³⁽²⁾) by the matrix 𝐽_∗𝜉²²,𝜉²⁵=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00𝜉0²²−1000100−𝜉₂₂00000000𝜉00²⁵−001²+1𝜉²⁵000010−0𝜉²²00⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝜉²²,𝜉²⁵∈ℝ,𝜉²⁵≠0.(4.1)𝐽_∗(𝜉²²,𝜉²⁵) is equivalent to 𝐽_∗(𝜉^22,𝜉^25) under some first (resp., second) kind automorphism if and only if 𝜉^22=𝜉²², 𝜉^25=𝜉²⁵ (resp., 𝜉^22=−𝜉²²,  𝜉^25=−((𝜉²²)²+1)/𝜉²⁵).

Proof. Let 𝐽=(𝜉^𝑖𝑗)_{1⩽𝑖,𝑗⩽6}=^𝐽1𝐽2_𝐽3𝐽4, (𝐽₁,𝐽₂,𝐽₃,𝐽₄3×3 blocks), an integrable complex structure in the basis (𝑌^ℓ(𝑘)). From Proposition 3.1, with some first kind automorphism, one may suppose 𝐽₁=^{00−10𝜉220100},  𝐽₄=^{00−10𝜉550100}. As Tr(𝐽)=0,  𝜉⁵⁵=−𝜉²². Then one is led to (4.1), and the result follows.

Remark 4.2. The complex subalgebra 𝔪 associated to 𝐽_∗(𝜉²²,𝜉²⁵) has basis 𝑌¹⁽¹⁾=𝑌¹⁽¹⁾−𝑖𝑌³⁽¹⁾, 𝑌¹⁽²⁾=𝑌¹⁽²⁾−𝑖𝑌³⁽²⁾, 𝑌²⁽²⁾=−𝑖𝜉²⁵𝑌²⁽¹⁾+(1+𝑖𝜉²²)𝑌²⁽²⁾. The complexification 𝔰𝔩(2)×𝔰𝔩(2) of 𝔰𝔩(2,ℝ)×𝔰𝔩(2,ℝ) has weight spaces decomposition with respect to the Cartan subalgebra 𝔥=ℂ𝑌²⁽¹⁾⊕ℂ𝑌²⁽²⁾: 𝔥⊕ℂ𝑌¹⁽¹⁾+𝑖𝑌³⁽¹⁾⊕ℂ𝑌¹⁽²⁾+𝑖𝑌³⁽²⁾⊕ℂ𝑌¹⁽¹⁾⊕ℂ𝑌¹⁽²⁾.(4.2) Then 𝔪=(𝔥∩𝔪)⊕ℂ𝑌¹⁽¹⁾⊕ℂ𝑌¹⁽²⁾ with 𝔥∩𝔪=ℂ𝑌²⁽²⁾, which is a special case of the general fact proved in [10] that any complex (integrable) structure on a reductive Lie group of class I is regular.

5. Case of 𝔫

Let 𝔫 be the real 3-dimensional Heisenberg Lie algebra with basis (𝑥₁,𝑥₂,𝑥₃) and commutation relations [𝑥₁,𝑥₂]=𝑥₃.

Proposition 5.1. Let 𝐽∶𝔫→𝔫 any linear map. 𝐽 has zero torsion if and only if it is equivalent to one of the endomorphisms defined in the basis (𝑥₁,𝑥₂,𝑥₃) by the following: (i)𝑆𝜉³³=⎛⎜⎜⎜⎝010−001𝜉00₃₃⎞⎟⎟⎟⎠,𝜉₃₃∈ℝ,(5.1)(ii)𝐷𝜉¹¹=⎛⎜⎜⎜⎜⎜⎝𝜉¹¹0𝜉0₁₁000𝜉₁₁0²−12𝜉¹¹⎞⎟⎟⎟⎟⎟⎠,𝜉₁₁∈ℝ,𝜉¹¹≠0,(5.2)(iii)𝑇(𝑎,𝑏)=⎛⎜⎜⎜⎜⎝010−0𝑏𝑎𝑏𝑎𝑏00−1𝑏⎞⎟⎟⎟⎟⎠,𝑎,𝑏∈ℝ,𝑏≠0.(5.3) Any two distinct endomorphisms in the preceding list are nonequivalent. 𝑇(𝑎,𝑏) is equivalent to 𝑇^(𝑎,𝑏)=⎛⎜⎜⎜⎜⎝𝑎0𝑏−00𝑏𝑎𝑏00−1𝑏⎞⎟⎟⎟⎟⎠.(5.4)

Proof. Let 𝐽=(𝜉^𝑖𝑗)_{1⩽𝑖,𝑗⩽3} in the basis (𝑥₁,𝑥₂,𝑥₃).The 9 torsion equations reduce to 𝜉¹³=𝜉²³=0 and equation 12∣3 (with the general notation introduced after Definition 2.6) which reads 𝜉³³Tr(𝐴)=det(𝐴)−1,(5.5) where 𝐴=^𝜉11𝜉12_𝜉21𝜉22. Suppose first Tr(𝐴)=0. Then 𝐴²=−𝐼, so that 𝐴 is similar over ℂ, hence over ℝ, to ⁰⁻¹¹⁰. Hence 𝐽≡^{0−10100∗∗𝜉}₃₃. Now, since 𝜉³³ does not belong to the spectrum of ⁰⁻¹¹⁰, taking the automorphism ^{100010𝛼𝛽1} of 𝔫 for suitable 𝛼,𝛽∈ℝ, one gets 𝐽≡𝑆(𝜉³³). Suppose now Tr(𝐴)≠0. Then 𝜉³³=(det(𝐴)−1)/Tr(𝐴). If 𝐴 is a scalar matrix, that is, 𝐴=𝜉¹¹𝐼, then 𝐽=^{𝜉11000𝜉110∗∗((𝜉}_{11)2−1)/2𝜉11}≡𝐷(𝜉¹¹). If 𝐴 is not a scalar matrix, then 𝐴 is similar to ^{0−𝑎𝑏1𝑏} for some 𝑎,𝑏∈ℝ, and 𝑏≠0 from the trace. Then 𝐽≡𝑇(𝑎,𝑏). Finally, 𝑇′(𝑎,𝑏)≡𝑇(𝑎,𝑏) since the matrices ^{0−𝑎𝑏1𝑏} and ^{𝑏−𝑏𝑎0} are similar for they have the same spectrum and are no scalar matrices.

Remark 5.2. 𝑆(𝜉³³) is an extension of a rank 1 𝐶𝑅-structure; however, 𝐷(𝜉¹¹),𝑇(𝑎,𝑏) are not.

6. 𝐶𝑅-Structures on 𝔫

Proposition 6.1. (i) Any linear map 𝐽∶𝔫→𝔫 which has zero torsion and is an extension of a rank 1 𝐶𝑅-structure on 𝔫 such that 𝔭 is nonabelian is equivalent to a unique ⎛⎜⎜⎜⎝010−001𝜉00₃₃⎞⎟⎟⎟⎠,𝜉₃₃∈ℝ.(6.1)
(ii) Any linear map 𝐽∶𝔫→𝔫 which is an extension of a rank 1 𝐶𝑅-structure on 𝔫 such that 𝔭 is abelian is equivalent to a unique ⎛⎜⎜⎜⎝𝜉00¹¹−001010⎞⎟⎟⎟⎠,𝜉₁₁∈ℝ.(6.2)𝐽 has nonzero torsion.

Proof. For any nonzero 𝑋∈𝔭, (𝑋,𝐽_𝔭𝑋) is a basis of 𝔭. In case (i), [𝑋,𝐽_𝔭𝑋]≠0, since 𝔭 is nonabelian. Then [𝑋,𝐽_𝔭𝑋]=𝜇𝑥₃, 𝜇≠0, and 𝑥₃∉𝔭 since otherwise 𝔭 would be abelian. One may extend 𝐽_𝔭 to 𝔫 in the basis (𝑋,𝐽_𝔭𝑋,𝜇𝑥₃) as 𝐽=⎛⎜⎜⎜⎝01−01𝜉𝜉¹³²³00𝜉³³⎞⎟⎟⎟⎠,(6.3) and 𝐽 has zero torsion only if 𝜉¹³=𝜉²³=0. In case (ii), necessarily 𝑥₃∈𝔭 since 𝔭 is abelian. Hence (𝑥₃,𝐽_𝔭𝑥₃) is a basis for 𝔭. Take any linear extension 𝐽 of 𝐽_𝔭 to 𝔫. There exists some eigenvector 𝑦₁≠0 of 𝐽 associated to some eigenvalue 𝜉¹¹∈ℝ. Then 𝑦₁∉𝔭, which implies [𝑦₁,𝐽𝑥₃]≠0, for otherwise 𝑦₁ would commute to the whole of 𝔫 and then be some multiple of 𝑥₃∈𝔭. Hence [𝑦₁,𝐽𝑥₃]=𝜆𝑥₃, 𝜆≠0, and dividing 𝑦₁ by 𝜆 one may suppose 𝜆=1. In the basis 𝑦₁,𝑦₂=𝐽𝑥₃,𝑦₃=𝑥₃ one has 𝐽=⎛⎜⎜⎜⎝𝜉¹¹00−001100⎞⎟⎟⎟⎠,(6.4) and (ii) follows.

Remark 6.2. Let 𝑁={[𝑥,𝑦,𝑧];𝑥,𝑦,𝑧∈ℝ} denote the Heisenberg group, where [𝑥,𝑦,𝑧]=^{1𝑥𝑧01𝑦001}. 𝑁 can also be realized ([11, 12]) as the boundary 𝑀₂={(𝜁,𝑤)∈ℂ²;ℑ𝜁=𝑤𝑤} of the Siegel half-space equipped with the multiplication (𝜁₁,𝑤₁)(𝜁₂,𝑤₂)=(𝜁₁+𝜁₂+2𝑖𝑤₁𝑤₂,𝑤₁+𝑤₂). The map Ψ∶𝑁→𝑀₂ defined by Ψ([𝑥,𝑦,𝑧])=(𝑧−(1/2)𝑥𝑦+𝑖(𝑥²+𝑦²)/4,(1/2)(𝑥−𝑖𝑦)) is an isomorphism. If 𝑃, 𝑄 denote the left invariant vector fields associated, respectively, to 𝑥₁,𝑥₂, then (𝑑Ψ)(𝑃+𝑖𝑄)=2𝑖𝑤(𝜕/𝜕𝜁)+(𝜕/𝜕𝑤), hence the left invariant 𝐶𝑅-structure on 𝑁 associated to the 𝐶𝑅-structure on 𝔫 introduced in (i) is the 𝐶𝑅-structure on 𝑀₂ induced from ℂ². The left invariant 𝐶𝑅-structure on 𝑁 associated to the 𝐶𝑅-structure on 𝔫 introduced in (ii) is not 𝐶𝑅-diffeomorphic to the 𝐶𝑅-structure on 𝑀₂ induced from ℂ², since the former has zero Levi form.

7. Complex Structures on 𝔫×𝔫

We will use for commutation relations [𝑥₁,𝑥₂]=𝑥₅,[𝑥₃,𝑥₄]=𝑥₆. The automorphisms of 𝔫×𝔫 fall into 2 kinds. The first kind is comprised of the matrices Φ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑏¹¹𝑏𝑏¹²0000₂₁0𝑏0²²𝑏0₃₃00000𝑏₄₃𝑏³⁴00𝑏⁴⁴00𝑏𝑏⁵¹₆₁𝑏⁵²𝑏⁵³𝑏⁵⁴𝑏¹¹𝑏²²−𝑏¹²𝑏²¹0𝑏⁶²𝑏⁶³𝑏⁶⁴0𝑏³³𝑏⁴⁴−𝑏³⁴𝑏⁴³⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝑏₁₁𝑏²²−𝑏¹²𝑏²¹𝑏³³𝑏⁴⁴−𝑏³⁴𝑏⁴³≠0.(7.1) The second kind ones are Ψ=ΘΦ where Φ is first kind and Θ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝001000011000010000000000000000000110⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(7.2)

Proposition 7.1. Any integrable complex structure 𝐽 on 𝔫×𝔫 is equivalent under some first kind automorphism to one of the following: (i)𝑆_𝜀𝜉⁵⁵=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝01000−0000100010−00001𝜉0000₅₅00000000𝜀−𝜀𝜉−⁵⁵𝜉²₅₅+1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝜀=±1,𝜉₅₅∈ℝ,(7.3)𝑆_𝜀′(𝜉^55) is equivalent to 𝑆_𝜀(𝜉⁵⁵) (𝜀,𝜀^=±1;𝜉^55,𝜉⁵⁵∈ℝ) under some first (resp., second) kind automorphism if and only if 𝜀^=𝜀,𝜉^55=𝜉⁵⁵ (resp.,  𝜀^=−𝜀,𝜉^55=−𝜉⁵⁵ ).(ii)𝐷𝜉¹¹=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜉¹¹0𝜉0₁₁−𝜉¹¹²+100010−0𝜉₁₁−𝜉¹¹²+100010−0𝜉₁₁000000𝜉₁₁00²−12𝜉¹¹−𝜉¹¹²+1²2𝜉¹¹000012𝜉¹¹1−𝜉¹¹²2𝜉¹¹⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝜉₁₁∈ℝ⧵{0},(7.4)𝐷(𝜉^11) is equivalent to 𝐷(𝜉¹¹) (𝜉^11,𝜉¹¹∈ℝ) under some first (resp., second) kind automorphism if and only if 𝜉^11=𝜉¹¹ (resp., 𝜉^11=−𝜉¹¹ ),(iii)𝑇𝜉³³,𝜉⁴³=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0−𝜉⁴³1−𝜉𝜉³³₃₃−𝜉⁴³𝜉³³𝜉⁴³𝜉³³−100−𝜉³³²+1−𝜉⁴³𝜉³³𝜉³³0𝜉₃₃𝜉³³0010𝜉𝜉³³₄₃−𝜉³³0000000−𝜉₄₃00𝜉³³−1𝜉³³−𝜉⁴³𝜉³³−2𝜉⁴³𝜉³³+𝜉³³²+1𝜉³³²00001𝜉⁴³𝜉³³−1𝜉³³𝜉₃₃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,∈ℝ⧵{0},𝜉⁴³∈ℝ,(7.5)𝑇(𝜉^33,𝜉^43) is equivalent to 𝑇(𝜉³³,𝜉⁴³) (𝜉^33,𝜉³³∈ℝ⧵{0},𝜉^43,𝜉⁴³∈ℝ) under some first (resp., second) kind automorphism if and only if 𝜉^33=𝜉³³,𝜉^43=𝜉⁴³ (resp.  𝜉^33=−𝜉³³,𝜉^43=−𝜉⁴³).
Finally, the cases (i), (ii), (iii) are mutually nonequivalent, either under first or second kind automorphism.

Proof. Let 𝐽=(𝜉^𝑖𝑗)_{1⩽𝑖,𝑗⩽6} an integrable complex structure in the basis (𝑥_𝑘)_1⩽𝑘⩽6. Denote 𝐽₁=^𝜉11𝜉12_𝜉21𝜉22, 𝐽₂=^𝜉13𝜉14_𝜉23𝜉24, 𝐽₃=^𝜉31𝜉32_𝜉41𝜉42, 𝐽₄=^𝜉33𝜉34_𝜉43𝜉44. Then 𝐽^∗1=^{𝜉11𝜉12𝜉15𝜉21𝜉22𝜉25}_{𝜉51𝜉52𝜉55} and 𝐽^∗3=^{𝜉33𝜉34𝜉36𝜉43𝜉44𝜉46}_{𝜉63𝜉64𝜉66} are zero torsion linear maps from 𝔫 to 𝔫, hence equivalent to type (5.1), (5.2), or (5.3) in Proposition 5.1. It can be checked that their being different types would contradict with 𝐽²=−1. Hence, modulo equivalence under some first kind automorphism, we get 3 cases: Case 1. 𝐽^∗1=^{0−1010000𝜉}₅₅, 𝐽^∗3=^{0−1010000𝜉}₆₆,Case 2. 𝐽^∗1=𝐷(𝜉¹¹),𝐽^∗3=𝐷(𝜉³³), (𝜉¹¹,𝜉³³≠0), andCase 3. 𝐽^∗1=^{0𝜉1201𝜉22000𝜉}₅₅, 𝐽^∗3=^{𝜉33−𝜉330𝜉430000𝜉}₆₆, (𝜉²²,𝜉³³≠0).Case 1 (resp., 2 and 3) leads to (7.3) (resp., (7.4), (7.5)). The assertion about equivalence in Cases 1 and 2 are readily proved, as is equivalence under some first kind automorphism in Case 3 and the nonequivalence of the 3 types. Consider now Θ𝑇(𝜉³³,𝜉⁴³)Θ⁻¹. It is equivalent under some first kind automorphism to some 𝑇(𝜂³³,𝜂⁴³). That implies that the matrices ^{𝜉33−𝜉33}_𝜉⁴³₀, ^{0−𝜂43𝜂33}_1−𝜂³³ are similar, which amounts to their having same trace and same determinant, that is, 𝜂³³=−𝜉³³,𝜂⁴³=−𝜉⁴³. As 𝑇(𝜉^33,𝜉^43) is equivalent to 𝑇(𝜉³³,𝜉⁴³) under some second kind automorphism if and only if it is equivalent to Θ𝑇(𝜉³³,𝜉⁴³)Θ⁻¹ under some first kind automorphism, the assertion about second kind equivalence in Case 3 follows.

Remark 7.2. In Case 3, had we used 𝐽^∗3=^{0𝜉3401𝜉44000𝜉}₆₆, then we would have to separate further into 2 subcases: subcase 𝜉¹²≠0, 𝑇𝜉¹²,𝜉²²=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0𝜉¹²−𝜉²²𝜉¹²1𝜉₂₂−𝜉¹²+100𝜉¹²+1𝜉¹²0−𝜉₁₂−𝜉²²001𝜉₂₂0𝜉¹²000001−0𝜉²²−𝜉₁₂00+1𝜉²²−𝜉²²²+𝜉¹²+1²𝜉²²𝜉¹²0000𝜉¹²𝜉²²𝜉¹²+1𝜉²²⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝜉₁₂𝜉²²≠0,(7.6) subcase 𝜉¹²=0, 𝑇𝜉²²=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝01𝜉0²²−1000−1𝜉₂₂0001000000−𝜉²²0²+101−0𝜉²²−010𝜉²²1𝜉²²0000−𝜉²²²+1𝜉²²1𝜉²²⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝜉₂₂≠0.(7.7)

Remark 7.3. 𝑆_𝜀(𝜉⁵⁵) is abelian (i.e., the corresponding complex subalgebra 𝔪 is abelian).

Remark 7.4. If one looks for zero torsion linear maps instead of complex structures, then 𝐽^∗1 and 𝐽^∗3 may be of different types.