# About Zero Torsion Linear Maps on Lie Algebras

Authors: Louis Magnin
Journal: ISRN Algebra
 (2011)

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## Abstract

We prove that any zero torsion linear map on a nonsolvable real Lie algebra𝔤0 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.


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## Body

## 1. Introduction

Given a real Lie algebra𝔤0, 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 andCR-structures on various constructions of Lie algebras, for example like those considered in [3] (see also [4]).


## 2. Zero Torsion Linear Maps and Extension ofCR-Structures

ACR-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.IfG0 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 rankr(1⩽r⩽[(dim𝔤0)/2])CR-structure on 𝔤0 is a r-dimensional complex subalgebra 𝔪 of 𝔤 such that 𝔪∩𝔪¯={0}.Ifdim𝔤0 is even, a CR-structure of maximal rank is an (integrable) complex structure.Now one has the following straightforward lemma.Lemma 2.2.
Let𝔭 be vector subspace of 𝔤0 and J𝔭:𝔭→𝔭 a linear map. Consider the real vector subspace 𝔪={X̃;X∈𝔭} of 𝔤, where X̃=X-iJ𝔭X for X∈𝔭. Denote 𝔪′=i𝔪={J𝔭X+iX;X∈𝔭}. Then (i)
𝔪 is a complex vector subspace of 𝔤 if and only if J𝔭2=-1𝔭;(ii)
𝔪 is stable with respect to the bracket if and only if for anyX,Y∈𝔭(2.1)[X,Y]-[JpX,JpY]∈p,Jp([X,Y]-[JpX,JpY])=[JpX,Y]+[X,JpY].
In that case, [X̃,Ỹ]=Z̃ with Z=[X,Y]-[J𝔭X,J𝔭Y].(iii)
𝔪′ is stable with respect to the bracket if and only if for any X,Y∈𝔭(2.2)[JpX,Y]+[X,JpY]∈p,Jp([JpX,Y]+[X,JpY])=[JpX,JpY]-[X,Y].
In that case, [X̃,Ỹ]=-iZ̃ with Z=[J𝔭X,Y]+[X,J𝔭Y] for X,Y∈𝔭.Remark 2.3.
When (i) is satisfied, (ii) and (iii) are trivially equivalent since then𝔪=𝔪′.Lemma 2.4.
A rankrCR-structure on 𝔤0 can be defined in an alternative way as (𝔭,J𝔭) where 𝔭 is a 2r-dimensional (1⩽r⩽[(dim𝔤0)/2]) vector subspace of 𝔤0 and J𝔭:𝔭→𝔭 is a linear map satisfying the 3 conditions
(2.3)Jp2=-1p,(2.4)[JpX,Y]+[X,JpY]∈p∀X,Y∈p,(2.5)[JpX,JpY]-[X,Y]-Jp([JpX,Y]+[X,JpY])=0∀X,Y∈p.Proof.
Let𝔪 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 J𝔭:𝔭→𝔤0 such that 𝔪={X̃=X-iJ𝔭X;X∈𝔭}. Now, for X∈𝔭,  iX̃=J𝔭X+iX∈i𝔪=𝔪 hence J𝔭X∈𝔭, so that J𝔭:𝔭→𝔭. Then (2.3), (2.4), (2.5) follow from Lemma 2.2 and Remark 2.3.
The converse comes again from Lemma2.2 and Remark 2.3.Remark 2.5.
The condition (2.3) implies detJ𝔭=1 and Trace (J𝔭)=0, hence if r=1, (2.4) and (2.5) follow from (2.3) and can be omitted.Definition 2.6.
A linear mapJ:𝔤0→𝔤0 is said to have zero torsion if it satisfies the condition
(2.6)[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]=0∀X,Y∈g0.IfJ 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 Ĵ associated to J (for more details, see [8]).When computing the matrices of the zero torsion maps in some fixed basis(xj)1⩽j⩽n of 𝔤0, we will denote by ij∣k (1⩽i,j,k⩽n) the torsion equation obtained by projecting on xk the equation (2.6) with X=xi, Y=xj.The automorphism groupAut  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,J𝔭)↦(Φ𝔭,Φ∘J𝔭∘Φ-1). Two J,J′ (resp., (𝔭,J𝔭), (𝔭′,J𝔭′)) are said to be equivalent (notation: J≡J′ (resp., J𝔭≡J𝔭′)) 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.
LetJ:𝔤0→𝔤0 be a linear map, 𝔪={X̃=X-iJX;X∈𝔤0} and 𝔪′=i𝔪={JX+iX;X∈𝔤0};(i)
𝔪∩𝔪′={X̃;X∈ker(J2+1)},(ii)
𝔪′ is a real subalgebra of 𝔤 if and only if J has zero torsion,(iii)
ifJ has zero torsion, 𝔪∩𝔪′ is a complex subalgebra of 𝔤.Proof.
(i) For anyZ∈𝔤 one has(2.7)Z∈m∩m′⟺∃X,Y∈g0,Z=X-iJX=JY+iY⟺∃X,Y∈g0,Z=X-iJX,X=JY,Y=-JX⟺∃X∈g0,Z=X-iJX,X=-J2X⟺∃X∈ker⁡(J2+1),Z=X̃.
(ii) The result follows from Lemma2.2(iii) since the first condition in (2.2) (with𝔭=𝔤0 and J𝔭=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.
LetJ: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 J𝔭 of J to 𝔭 is a CR-structure on 𝔤0.Definition 2.9.
A real form𝔲 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 projectionsπ1: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.
LetJ:𝔤0→𝔤0 be a zero torsion linear map, 𝔪={X̃=X-iJX;X∈𝔤0} and 𝔪′=i𝔪.  J is an extension of a CR-structure if and only if 𝔪∩𝔪′≠{0}.Proof.
From Lemma2.7, 𝔪∩𝔪′ is a complex subalgebra of 𝔤 and 𝔪∩𝔪′={X̃;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 J𝔭 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,  X̃,Ỹ∈𝔪∩𝔪′ hence [iX̃,Ỹ]=[J𝔭X+iX,Y-iJ𝔭Y]∈𝔪∩𝔪′ and (2.4) follows.Corollary 2.12.
There is a one-to-one correspondence between nontype I real forms of𝔤 and zero torsion linear maps J:𝔤0→𝔤0 which are no extension of a CR-structure.Proof.
LetJ: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.
If𝔤0 is of type I, then any zero torsion linear map J:𝔤0→𝔤0 is an extension of a CR-structure.Proposition 2.14.
Let𝔲 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.
LetZ 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.
If𝔤0 is nonsolvable, then it is type I.Proof.
If𝔤0 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 of𝔰𝔩(2,ℝ)

G=SL(2,ℝ) denotes the Lie group of real 2×2 matrices with determinant 1(3.1)σ=(abcd),ad-bc=1.
Its Lie algebra 𝔤0=𝔰𝔩(2,ℝ) consists of the zero trace real 2×2 matrices (3.2)X=(xyz-x)=xH+yX++zX-,
with basis H=(100-1), X+=(0100), X-=(0010) and commutation relations(3.3)[H,X+]=2X+,[H,X-]=-2X-,[X+,X-]=H.
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(3.4)[Y1,Y2]=Y3,[Y1,Y3]=Y2,[Y2,Y3]=Y1.
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-) is(3.5)Φ=(1+2bc-acbd-2aba2-b22cd-c2d2).
The adjoint group Ad(G) is the identity component of Aut  𝔤0 and one has(3.6)Autg0=Ad(G)∪Ψ0Ad(G),Ψ0=diag⁡(1,-1,-1).
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 sheetz>0 of the cone x2+yz=0 (orbit of X-);(iii)
the lower sheetz<0 of the cone x2+yz=0 (orbit of -X-);(iv)
for alls>0 the one-sheet hyperboloid x2+yz=s2 (orbit of sH);(v)
for alls>0 the upper sheet z>0 of the hyperboloid x2+yz=-s2 (orbit of s(-X++X-));(vi)
for alls>0 the lower sheet z<0 of the hyperboloid x2+yz=-s2 (orbit of s(X+-X-)).The orbits of𝔤0 under the whole Aut𝔤0 are as follows, beside {0}: (I)
the conex2+yz=0 (orbit of X-);(II)
the one-sheet hyperboloidx2+yz=s2 (orbit of sH) (s>0);(III)
the two-sheet hyperboloidx2+yz=-s2 (orbit of s(X+-X-)) (s>0).Proposition 3.1.
Let𝔤0=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
(3.7)J*(λ)=(00-10λ0100),λ∈R,J*(λ)≢J*(μ) for λ≠μ (resp.,
(3.8)J(α)=(0-12-121α-α1-αα),α∈R,J(α)≢J(β) for α≠β).Proof.
LetJ=(ξji)1⩽i,j⩽3 in the basis (H,X+,X-). The 9 torsion equations are in the basis (H,X+,X-):
(3.9)12∣12(ξ22+ξ11)ξ21+(ξ22-ξ11)ξ13-(ξ12+2ξ31)ξ23=0,12∣22(ξ12ξ21+1+(ξ22)2)-ξ13ξ12-2ξ23ξ32=0,12∣3(ξ13+2ξ21)ξ13-2(ξ22+2ξ11)ξ23+2ξ33ξ23=0,13∣1(ξ12-2ξ31)ξ11+2ξ32ξ21+ξ13ξ32-(ξ12+2ξ31)ξ33=0,13∣22(ξ22-2ξ11)ξ32+(ξ12+2ξ31)ξ12-2ξ33ξ32=0,13∣3ξ13ξ12-2ξ13ξ31-2+2ξ23ξ32-2(ξ33)2=0,23∣14ξ31ξ21-1-ξ22ξ11-ξ23ξ32+(ξ22-ξ11)ξ33=0,23∣24ξ32ξ21-(ξ22+ξ33)ξ12=0,23∣34ξ23ξ31-(ξ22+ξ33)ξ13=0.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 existsφ∈Aut𝔤0 such that v=φ(X-). Then, replacing J by φ-1Jφ, we may suppose ξ31=ξ32=0. That case is impossible from 13∣2 and 13∣3.Case 2.
There existsφ∈Aut𝔤0 such that v=φ(H). Then we may suppose ξ12=ξ13=0. Then from 12∣2, ξ32ξ23≠0, and 23∣2, 23∣3 yield ξ21=ξ31=0. Then 12∣3 and 13∣2 successively give ξ33=ξ22+2ξ11 and ξ11=0. Now 12∣2 and 23∣1 read respectively -ξ32ξ23+(ξ22)2+1=0, and ξ32ξ23-(ξ22)2+1=0. Hence that case is impossible.Case 3.
There existsφ∈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 form(3.10)J*=(η110η31η12λη32η130η32).
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: (3.11)*12∣1(η13+η31)λ-(η13-η31)η11=0,*12∣2(η11+λ)η32-η12η13=0,*12∣3η11λ-1+(η13)2-(η11+λ)η33=0,*13∣1η32η31+η12η11+η32η13-η12η33=0,*13∣2η11λ+1+(η12)2+(η32)2+η13η31-(η11-λ)η33=0,*13∣3η32η11-η12(η31+η13)-η32η33=0,*23∣1η11λ+1-(η31)2+(η11-λ)η33=0,*23∣2η32η31-(η33+λ)η12=0,*23∣3(η13+η31)λ+(η13-η31)η33=0.
From *12∣1 and *23∣3, (3.12)η11(η13-η31)=-η33(η13-η31).Case 1.
Suppose first thatη13=η31. Then λη13=0.Subcase 1.
Consider the subcaseη13=0.*13∣1 and *13∣3 read, respectively, (η33-η11)η12=0,(η33-η11)η32=0. Suppose η33≠η11. Then η12=η32=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 Subcase2:η13≠0. Then λ=0. From *13∣2, η33η11≠0. Then *23∣1 yields η33=(-1+(η13)2)/η11 and *13∣2 reads (η12)2+(η32)2+2=0. This Subcase 2 is not possible either.Case 2.
Hence Case1 is not possible, and we are necessarily in the Case 2:η13≠η31. From (3.12), η33=-η11. Then *13∣2 reads (η11)2+(η12)2+(η32)2+1+η13η31=0 hence η13≠0 and η31=-((η11)2+(η12)2+(η32)2+1)/η13. From *12∣2,η12=(η32(η11+λ))/η13. Then *23∣2 reads η32(((η32)2+λ2+1)(η13)2+(η11+λ)2(η32)2)=0, that is, η32=0, which implies η12=0. Now *12∣1 reads λ(1+(η11)2-(η13)2)=-η11(1+(η11)2+(η13)2). The subcase η11≠0 is not possible since then *12∣3 would yield λ=-((η11)2+(η13)2-1)/2η11 and *12∣1 would read ((η11)2+(η13+1)2)((η11)2+(η13-1)2)=0. Hence η11=0. Then *12∣3 reads (ξ13)2=1. The condition (ξ13)2=1 now implies the vanishing of all the torsion equations. In that case
(3.13)J*=(00-ɛ0λ0ɛ00),ɛ=±1.
Then in the basis (H,X+,X-)(3.14)J=(0-ɛ2-ɛ2ɛλ2-λ2ɛ-λ2λ2).
The cases ɛ=±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 on𝔰𝔩(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 structureJ 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
(4.1)J̃*(ξ22,ξ52)=(00-10000ξ2200ξ52010000000000-10-(ξ22)2+1ξ5200-ξ220000100),ξ22,ξ52∈R,ξ52≠0.J̃*(ξ22,ξ52) is equivalent to J̃*(ξ′22,ξ′52) under some first (resp., second) kind automorphism if and only if ξ′22=ξ22, ξ′52=ξ52 (resp., ξ′22=-ξ22,  ξ′52=-((ξ22)2+1)/ξ52).Proof.
LetJ=(ξji)1⩽i,j⩽6=(J1J2J3J4), (J1,J2,J3,J43×3 blocks), an integrable complex structure in the basis (Yℓ(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 subalgebra𝔪 associated to J̃*(ξ22,ξ52) has basis Ỹ1(1)=Y1(1)-iY3(1), Ỹ1(2)=Y1(2)-iY3(2), Ỹ2(2)=-iξ52Y2(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):
(4.2)h⊕C(Y1(1)+iY3(1))⊕C(Y1(2)+iY3(2))⊕CỸ1(1)⊕CỸ1(2).
Then 𝔪=(𝔥∩𝔪)⊕ℂỸ1(1)⊕ℂỸ1(2) with 𝔥∩𝔪=ℂỸ2(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 of𝔫

Let𝔫 be the real 3-dimensional Heisenberg Lie algebra with basis (x1,x2,x3) and commutation relations [x1,x2]=x3.Proposition 5.1.
LetJ: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)
(5.1)S(ξ33)=(0-1010000ξ33),ξ33∈R,(ii)
(5.2)D(ξ11)=(ξ11000ξ11000(ξ11)2-12ξ11),ξ11∈R,ξ11≠0,(iii)
(5.3)T(a,b)=(0-ab01b000ab-1b),a,b∈R,b≠0.
Any two distinct endomorphisms in the preceding list are nonequivalent. T(a,b) is equivalent to
(5.4)T′(a,b)=(b-b0a0000ab-1b).Proof.
LetJ=(ξji)1⩽i,j⩽3 in the basis (x1,x2,x3).The 9 torsion equations reduce to ξ31=ξ32=0 and equation 12∣3 (with the general notation introduced after Definition 2.6) which reads
(5.5)ξ33Tr⁡(A)=det⁡(A)-1,
where A=(ξ11ξ21ξ12ξ22). 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**ξ33). 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**((ξ11)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 mapJ: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
(6.1)(0-1010000ξ33),ξ33∈R.
(ii) Any linear mapJ:𝔫→𝔫 which is an extension of a rank 1 CR-structure on 𝔫 such that 𝔭 is abelian is equivalent to a unique
(6.2)(ξ11000010-10),ξ11∈R.J has nonzero torsion.Proof.
For any nonzeroX∈𝔭, (X,J𝔭X) is a basis of 𝔭. In case (i), [X,J𝔭X]≠0, since 𝔭 is nonabelian. Then [X,J𝔭X]=μx3, μ≠0, and x3∉𝔭 since otherwise 𝔭 would be abelian. One may extend J𝔭 to 𝔫 in the basis (X,J𝔭X,μx3) as
(6.3)J=(0-1ξ3110ξ3200ξ33),
and J has zero torsion only if ξ31=ξ32=0. In case (ii), necessarily x3∈𝔭 since 𝔭 is abelian. Hence (x3,J𝔭x3) is a basis for 𝔭. Take any linear extension J of J𝔭 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
(6.4)J=(ξ11000010-10),
and (ii) follows.Remark 6.2.
LetN={[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+2iw1¯w2,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 on𝔫×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 (7.1)Φ=(b11b210000b12b22000000b33b430000b34b4400b15b25b35b45b11b22-b21b120b16b26b36b460b33b44-b43b34),(b11b22-b21b12)(b33b44-b43b34)≠0.
The second kind ones are Ψ=ΘΦ where Φ is first kind and (7.2)Θ=(001000000100100000010000000001000010).Proposition 7.1.
Any integrable complex structureJ on 𝔫×𝔫 is equivalent under some first kind automorphism to one of the following:(i)
(7.3)S̃ɛ(ξ55)=(0-10000100000000-1000010000000ξ55-ɛ((ξ55)2+1)0000ɛ-ξ55),ɛ=±1,ξ55∈R,S̃ɛ′(ξ′55) is equivalent to S̃ɛ(ξ55) (ɛ,ɛ′=±1;ξ′55,ξ55∈ℝ) under some first (resp., second) kind automorphism if and only if ɛ′=ɛ,ξ′55=ξ55 (resp.,  ɛ′=-ɛ,ξ′55=-ξ55 ).(ii)
(7.4)D̃(ξ11)=(ξ110-((ξ11)2+1)0000ξ110-((ξ11)2+1)0010-ξ11000010-ξ11000000(ξ11)2-12ξ11-((ξ11)2+1)22ξ11000012ξ111-(ξ11)22ξ11),ξ11∈R∖{0},D̃(ξ′11) is equivalent to D̃(ξ11) (ξ′11,ξ11∈ℝ) under some first (resp., second) kind automorphism if and only if ξ′11=ξ11 (resp., ξ′11=-ξ11 ),(iii)
(7.5)T̃(ξ33,ξ34)=(0-ξ34ξ33-ξ34ξ33ξ34ξ33-1001-ξ33-(ξ33)2+1-ξ34ξ33ξ33ξ33000ξ33ξ33-ξ330010ξ340000000-ξ34ξ33-1ξ33-(ξ34ξ33-2)ξ34ξ33+(ξ33)2+1(ξ33)200001ξ34ξ33-1ξ33),ξ33∈R∖{0},ξ34∈R,T̃(ξ′33,ξ′34) is equivalent to T̃(ξ33,ξ34) (ξ′33,ξ33∈ℝ∖{0},ξ′34,ξ34∈ℝ) under some first (resp., second) kind automorphism if and only if ξ′33=ξ33,ξ′34=ξ34 (resp.  ξ′33=-ξ33,ξ′34=-ξ34).
Finally, the cases (i), (ii), (iii) are mutually nonequivalent, either under first or second kind automorphism.Proof.
LetJ=(ξji)1⩽i,j⩽6 an integrable complex structure in the basis (xk)1⩽k⩽6. Denote J1=(ξ11ξ21ξ12ξ22), J2=(ξ31ξ41ξ32ξ42), J3=(ξ13ξ23ξ14ξ24), J4=(ξ33ξ43ξ34ξ44). Then J1*=(ξ11ξ21ξ51ξ12ξ22ξ52ξ15ξ25ξ55) and J3*=(ξ33ξ43ξ63ξ34ξ44ξ64ξ36ξ46ξ66) 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.
J1*=(0-1010000ξ55), J3*=(0-1010000ξ66),Case 2.
J1*=D(ξ11),J3*=D(ξ33), (ξ11,ξ33≠0), andCase 3.
J1*=(0ξ2101ξ22000ξ55), J3*=(ξ33-ξ330ξ340000ξ66), (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 ΘT̃(ξ33,ξ34)Θ-1. It is equivalent under some first kind automorphism to some T̃(η33,η34). That implies that the matrices (ξ33-ξ33ξ340), (0-η34η331-η33) are similar, which amounts to their having same trace and same determinant, that is, η33=-ξ33,η34=-ξ34. As T̃(ξ′33,ξ′34) is equivalent to T̃(ξ33,ξ34) under some second kind automorphism if and only if it is equivalent to ΘT̃(ξ33,ξ34)Θ-1 under some first kind automorphism, the assertion about second kind equivalence in Case 3 follows.Remark 7.2.
In Case3, had we used J3*=(0ξ4301ξ44000ξ66), then we would have to separate further into 2 subcases: subcase ξ21≠0,
(7.6)T̃(ξ21,ξ22)=(0ξ21-ξ22ξ21-(ξ21+1)001ξ22ξ21+1ξ21-ξ22000-ξ210ξ21001ξ221-ξ22000000-ξ21+1ξ22-(ξ22)2+(ξ21+1)2ξ22ξ210000ξ21ξ22ξ21+1ξ22),ξ21ξ22≠0,
subcase ξ21=0,
(7.7)T̃(ξ22)=(00-10001ξ220100100000-ξ22-((ξ22)2+1)1-ξ22000000-1ξ221ξ220000-(ξ22)2+1ξ221ξ22),ξ22≠0.Remark 7.3.
S̃ɛ(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, thenJ1* and J3* may be of different types.



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*Source: 930189-2011-07-17.xml*