International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 930189 | https://doi.org/10.5402/2011/930189

Louis Magnin, "About Zero Torsion Linear Maps on Lie Algebras", International Scholarly Research Notices, vol. 2011, Article ID 930189, 16 pages, 2011. https://doi.org/10.5402/2011/930189

About Zero Torsion Linear Maps on Lie Algebras

Academic Editor: A. Kiliçman
Received06 May 2011
Accepted31 May 2011
Published17 Jul 2011

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, ) 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,) 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 𝑇(𝑀) of 𝑀 such that [𝒱,𝒱]𝒱 (i.e., the space of smooth sections of 𝒱 is closed under commutators) and 𝒱𝒱={0} ( denoting here conjugation in 𝑇(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(1𝑟[(dim𝔤0)/2])𝐶𝑅-structure on 𝔤0 is a 𝑟-dimensional complex subalgebra 𝔪 of 𝔤 such that 𝔪𝔪={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 𝐽𝔭𝔭𝔭 a linear map. Consider the real vector subspace 𝔪={𝑋;𝑋𝔭} of 𝔤, where 𝑋=𝑋𝑖𝐽𝔭𝑋 for 𝑋𝔭. Denote 𝔪=𝑖𝔪={𝐽𝔭𝑋+𝑖𝑋;𝑋𝔭}. Then (i)𝔪 is a complex vector subspace of 𝔤 if and only if 𝐽2𝔭=1𝔭;(ii)𝔪 is stable with respect to the bracket if and only if for any𝑋,𝑌𝔭[]𝐽𝑋,𝑌𝔭𝑋,𝐽𝔭𝑌𝐽𝔭,𝔭[]𝐽𝑋,𝑌𝔭𝑋,𝐽𝔭𝑌=𝐽𝔭+𝑋,𝑌𝑋,𝐽𝔭𝑌.(2.1) In that case, [𝑍𝑋,𝑌]= with 𝑍=[𝑋,𝑌][𝐽𝔭𝑋,𝐽𝔭𝑌].(iii)𝔪 is stable with respect to the bracket if and only if for any 𝑋,𝑌𝔭𝐽𝔭+𝑋,𝑌𝑋,𝐽𝔭𝑌𝐽𝔭,𝔭𝐽𝔭+𝑋,𝑌𝑋,𝐽𝔭𝑌=𝐽𝔭𝑋,𝐽𝔭𝑌[].𝑋,𝑌(2.2) In that case, [𝑍𝑋,𝑌]=𝑖 with 𝑍=[𝐽𝔭𝑋,𝑌]+[𝑋,𝐽𝔭𝑌] for 𝑋,𝑌𝔭.

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,𝐽𝔭) where 𝔭 is a 2𝑟-dimensional (1𝑟[(dim𝔤0)/2]) vector subspace of 𝔤0 and 𝐽𝔭𝔭𝔭 is a linear map satisfying the 3 conditions 𝐽2𝔭=1𝔭,𝐽(2.3)𝔭+𝑋,𝑌𝑋,𝐽𝔭𝑌𝐽𝔭𝑋,𝑌𝔭,(2.4)𝔭𝑋,𝐽𝔭𝑌[]𝑋,𝑌𝐽𝔭𝐽𝔭+𝑋,𝑌𝑋,𝐽𝔭𝑌=0𝑋,𝑌𝔭.(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=𝔪,  dim𝔭=2𝑟, and there exists a unique linear map 𝐽𝔭𝔭𝔤0 such that 𝔪={𝑋=𝑋𝑖𝐽𝔭𝑋;𝑋𝔭}. Now, for 𝑋𝔭,  𝑖𝑋=𝐽𝔭𝑋+𝑖𝑋𝑖𝔪=𝔪 hence 𝐽𝔭𝑋𝔭, so that 𝐽𝔭𝔭𝔭. 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 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 map 𝐽𝔤0𝔤0 is said to have zero torsion if it satisfies the condition [][][][]𝐽𝑋,𝐽𝑌𝑋,𝑌𝐽𝐽𝑋,𝑌𝐽𝑋,𝐽𝑌=0𝑋,𝑌𝔤0.(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 (𝑥𝑗)1𝑗𝑛 of 𝔤0, we will denote by 𝑖𝑗𝑘 (1𝑖,𝑗,𝑘𝑛) the torsion equation obtained by projecting on 𝑥𝑘 the equation (2.6) with 𝑋=𝑥𝑖, 𝑌=𝑥𝑗.

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 𝐽Φ𝐽Φ1 for all ΦAut  𝔤0. It acts also on the set of 𝐶𝑅-structures by (𝔭,𝐽𝔭)(Φ𝔭,Φ𝐽𝔭Φ1). Two 𝐽,𝐽 (resp., (𝔭,𝐽𝔭), (𝔭,𝐽𝔭)) are said to be equivalent (notation: 𝐽𝐽 (resp., 𝐽𝔭𝐽𝔭)) 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 𝐽𝔤0𝔤0 be a linear map, 𝔪={𝑋=𝑋𝑖𝐽𝑋;𝑋𝔤0} and 𝔪=𝑖𝔪={𝐽𝑋+𝑖𝑋;𝑋𝔤0}; (i)𝔪𝔪={𝑋;𝑋ker(𝐽2+1)}, (ii)𝔪 is a real subalgebra of 𝔤 if and only if 𝐽 has zero torsion,(iii)if 𝐽 has zero torsion, 𝔪𝔪 is a complex subalgebra of 𝔤.

Proof. (i) For any 𝑍𝔤 one has𝑍𝔪𝔪𝑋,𝑌𝔤0,𝑍=𝑋𝑖𝐽𝑋=𝐽𝑌+𝑖𝑌𝑋,𝑌𝔤0,𝑍=𝑋𝑖𝐽𝑋,𝑋=𝐽Y,𝑌=𝐽𝑋𝑋𝔤0,𝑍=𝑋𝑖𝐽𝑋,𝑋=𝐽2𝑋𝐽𝑋ker2+1,𝑍=𝑋.(2.7)
(ii) The result follows from Lemma 2.2(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 [𝔪𝔪,𝔪𝔪]𝔪𝔪. Clearly 𝔪𝔪 is stable by multiplication by 𝑖.

Definition 2.8. Let 𝐽𝔤0𝔤0 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 form 𝔲 of 𝔤 is said to be of type I (with respect to 𝔤0) if 𝔤0𝔲{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 𝜋1𝔤𝔤0, 𝜋2𝔤𝔤0 defined by 𝑧=𝜋1(𝑧)+𝑖𝜋2(𝑧) for 𝑧𝔤. Then a real form 𝔲 of 𝔤 is of type I if and only if ker𝜋2𝔲{0}.

Proposition 2.11. Let 𝐽𝔤0𝔤0 be a zero torsion linear map, 𝔪={𝑋=𝑋𝑖𝐽𝑋;𝑋𝔤0} and 𝔪=𝑖𝔪.  𝐽 is an extension of a 𝐶𝑅-structure if and only if 𝔪𝔪{0}.

Proof. From Lemma 2.7, 𝔪𝔪 is a complex subalgebra of 𝔤 and 𝔪𝔪={𝑋;𝑋ker(𝐽2+1)}. If 𝐽 is an extension of a 𝐶𝑅-structure, one has {0}𝔭ker(𝐽2+1) hence 𝔪𝔪{0}. Conversely, if 𝔪𝔪{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), 𝔪𝔪 is a subalgebra of 𝔤. Precisely, for 𝑋,𝑌𝔭,  𝑋,𝑌𝔪𝔪 hence [𝑖𝑋,𝑌]=[𝐽𝔭𝑋+𝑖𝑋,𝑌𝑖𝐽𝔭𝑌]𝔪𝔪 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 𝐽𝔤0𝔤0 which are no extension of a 𝐶𝑅-structure.

Proof. Let 𝐽𝔤0𝔤0 be a zero torsion linear map that is no extension of a 𝐶𝑅-structure. Then 𝔪𝔪={0}, hence 𝔪={𝐽𝑋+𝑖𝑋;𝑋𝔤0} 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;𝑋𝔤0} for some linear map 𝐽𝔤0𝔤0, that is, 𝔲=𝔪. As 𝔪 is a real subalgebra, 𝐽 has zero torsion from Lemma 2.7(ii). Now 𝔪𝔪={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 𝐽𝔤0𝔤0 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=𝜎𝜏Aut𝔤. 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 𝑊𝔤0𝔲. Now, 𝑊=0 if and only if 𝜎𝑍=𝜏𝑍=𝑍, that is, 𝑖𝑍𝔤0𝔲.

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,)

𝐺=𝑆𝐿(2,) denotes the Lie group of real 2×2 matrices with determinant 1𝜎=𝑎𝑏𝑐𝑑,𝑎𝑑𝑏𝑐=1.(3.1) Its Lie algebra 𝔤0=𝔰𝔩(2,) consists of the zero trace real 2×2 matrices 𝑋=𝑥𝑦𝑧𝑥=𝑥𝐻+𝑦𝑋++𝑧𝑋,(3.2) with basis 𝐻=1001, 𝑋+=0100, 𝑋=0010 and commutation relations𝐻,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 relations𝑌1,𝑌2=𝑌3,𝑌1,𝑌3=𝑌2,𝑌2,𝑌3=𝑌1.(3.4) The adjoint representation of 𝐺 on 𝔤0 is given by Ad(𝜎)𝑋=𝜎𝑋𝜎1. The matrix Φ of Ad(𝜎) (𝜎 as in (3.1)) in the basis (𝐻,𝑋+,𝑋) isΦ=1+2𝑏𝑐𝑎𝑐𝑏𝑑2𝑎𝑏𝑎2𝑏22𝑐𝑑𝑐2𝑑2.(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,), and 𝐽𝔤0𝔤0 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)=0010𝜆0100,𝜆,(3.7)𝐽(𝜆)𝐽(𝜇) for 𝜆𝜇 (resp., 1𝐽(𝛼)=02121𝛼𝛼1𝛼𝛼,𝛼,(3.8)𝐽(𝛼)𝐽(𝛽) for 𝛼𝛽).

Proof. Let 𝐽=(𝜉𝑖𝑗)1𝑖,𝑗3 in the basis (𝐻,𝑋+,𝑋). The 9 torsion equations are in the basis (𝐻,𝑋+,𝑋): 𝜉121222+𝜉11𝜉12+𝜉22𝜉11𝜉31𝜉21+2𝜉13𝜉32𝜉=0,122221𝜉12𝜉+1+222𝜉31𝜉212𝜉32𝜉23𝜉=0,12331+2𝜉12𝜉31𝜉222+2𝜉11𝜉32+2𝜉33𝜉32𝜉=0,131212𝜉13𝜉11+2𝜉23𝜉12+𝜉31𝜉23𝜉21+2𝜉13𝜉33𝜉=0,1322222𝜉11𝜉23+𝜉21+2𝜉13𝜉212𝜉33𝜉23=0,133𝜉31𝜉212𝜉31𝜉132+2𝜉32𝜉23𝜉2332=0,2314𝜉13𝜉121𝜉22𝜉11𝜉32𝜉23+𝜉22𝜉11𝜉33=0,2324𝜉23𝜉12𝜉22+𝜉33𝜉21=0,2334𝜉32𝜉13𝜉22+𝜉33𝜉31=0.(3.9)𝐽 has at least one real eigenvalue 𝜆. Let 𝑣𝔤0,  𝑣0, 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 𝜑Aut𝔤0 such that 𝑣=𝜑(𝑋). Then, replacing 𝐽 by 𝜑1𝐽𝜑, we may suppose 𝜉13=𝜉23=0. That case is impossible from 132 and 133.Case 2. There exists 𝜑Aut𝔤0 such that 𝑣=𝜑(𝐻). Then we may suppose 𝜉21=𝜉31=0. Then from 122, 𝜉23𝜉320, and 232, 233 yield 𝜉12=𝜉13=0. Then 123 and 132 successively give 𝜉33=𝜉22+2𝜉11 and 𝜉11=0. Now 122 and 231 read respectively 𝜉23𝜉32+(𝜉22)2+1=0, and 𝜉23𝜉32(𝜉22)2+1=0. Hence that case is impossible.Case 3. There exists 𝜑Aut𝔤0 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 𝑖𝑗𝑘 (the star is to underline that the new basis is in use) for 𝐽 in that basis are as follows: 𝜂12131+𝜂13𝜂𝜆31𝜂13𝜂11𝜂=0,12211𝜂+𝜆23𝜂21𝜂31=0,123𝜂11𝜂𝜆1+312𝜂11𝜂+𝜆33=0,131𝜂23𝜂13+𝜂21𝜂11+𝜂23𝜂31𝜂21𝜂33=0,132𝜂11𝜂𝜆+1+212+𝜂232+𝜂31𝜂13𝜂11𝜂𝜆33=0,133𝜂23𝜂11𝜂21𝜂13+𝜂31𝜂23𝜂33=0,231𝜂11𝜂𝜆+1132+𝜂11𝜂𝜆33=0,232𝜂23𝜂13𝜂33𝜂+𝜆21𝜂=0,23331+𝜂13𝜂𝜆+31𝜂13𝜂33=0.(3.11) From 121 and 233, 𝜂11𝜂31𝜂13=𝜂33𝜂31𝜂13.(3.12)Case 1. Suppose first that 𝜂31=𝜂13. Then 𝜆𝜂31=0.Subcase 1. Consider the subcase 𝜂31=0.131 and 133 read, respectively, (𝜂33𝜂11)𝜂21=0,(𝜂33𝜂11)𝜂23=0. Suppose 𝜂33𝜂11. Then 𝜂21=𝜂23=0, and 132 gives 𝜂11𝜆+1=(𝜂11𝜆)𝜂33, which implies 𝜂33=0 by 231. As 123 then reads 1=0, this case 𝜂33𝜂11 is not possible. Now, the case 𝜂33=𝜂11 is not possible either since then 231 would read (𝜂11)2+1=0. We conclude that the Subcase 1 is not possible. Subcase 2. Hence we are in the Subcase 2:𝜂310. Then 𝜆=0. From 132, 𝜂33𝜂110. Then 231 yields 𝜂33=(1+(𝜂31)2)/𝜂11 and 132 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:𝜂31𝜂13. From (3.12), 𝜂33=𝜂11. Then 132 reads (𝜂11)2+(𝜂21)2+(𝜂23)2+1+𝜂31𝜂13=0 hence 𝜂310 and 𝜂13=((𝜂11)2+(𝜂21)2+(𝜂23)2+1)/𝜂31. From 122,𝜂21=(𝜂23(𝜂11+𝜆))/𝜂31. Then 232 reads 𝜂23(((𝜂23)2+𝜆2+1)(𝜂31)2+(𝜂11+𝜆)2(𝜂23)2)=0, that is, 𝜂23=0, which implies 𝜂21=0. Now 121 reads 𝜆(1+(𝜂11)2(𝜂31)2)=𝜂11(1+(𝜂11)2+(𝜂31)2). The subcase 𝜂110 is not possible since then 123 would yield 𝜆=((𝜂11)2+(𝜂31)21)/2𝜂11 and 121 would read ((𝜂11)2+(𝜂31+1)2)((𝜂11)2+(𝜂311)2)=0. Hence 𝜂11=0. Then 123 reads (𝜉31)2=1. The condition (𝜉31)2=1 now implies the vanishing of all the torsion equations. In that case 𝐽=00𝜀0𝜆0𝜀00,𝜀=±1.(3.13) Then in the basis (𝐻,𝑋+,𝑋)𝜀𝐽=02𝜀2𝜀𝜆2𝜆2𝜆𝜀2𝜆2.(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,)×𝔰𝔩(2,)

We consider the basis (𝑌1(1),𝑌2(1),𝑌3(1),𝑌1(2),𝑌2(2),𝑌3(2)) of 𝔰𝔩(2,)×𝔰𝔩(2,), with the upper index referring to the first or second factor. The automorphisms of 𝔰𝔩(2,)×𝔰𝔩(2,) fall into 2 kinds: the first kind is comprised of the diag(Φ1,Φ2), Φ1,Φ2Aut𝔰𝔩(2,), and the second kind is comprised of the Γdiag(Φ1,Φ2), with Γ the switch between the two factors of 𝔰𝔩(2,)×𝔰𝔩(2,).

Proposition 4.1. Any integrable complex structure 𝐽 on 𝔰𝔩(2,)×𝔰𝔩(2,) 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 𝐽𝜉22,𝜉25=0010000𝜉2200𝜉250𝜉1000000000010222+1𝜉2500𝜉220000100,𝜉22,𝜉25,𝜉250.(4.1)𝐽(𝜉22,𝜉25) is equivalent to 𝐽(𝜉22,𝜉25) under some first (resp., second) kind automorphism if and only if 𝜉22=𝜉22, 𝜉25=𝜉25 (resp., 𝜉22=𝜉22,  𝜉25=((𝜉22)2+1)/𝜉25).

Proof. Let 𝐽=(𝜉𝑖𝑗)1𝑖,𝑗6=𝐽1𝐽2𝐽3𝐽4, (𝐽1,𝐽2,𝐽3,𝐽43×3 blocks), an integrable complex structure in the basis (𝑌(𝑘)). From Proposition 3.1, with some first kind automorphism, one may suppose 𝐽1=0010𝜉220100,  𝐽4=0010𝜉550100. 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 𝑌1(1)=𝑌1(1)𝑖𝑌3(1), 𝑌1(2)=𝑌1(2)𝑖𝑌3(2), 𝑌2(2)=𝑖𝜉25𝑌2(1)+(1+𝑖𝜉22)𝑌2(2). The complexification 𝔰𝔩(2)×𝔰𝔩(2) of 𝔰𝔩(2,)×𝔰𝔩(2,) has weight spaces decomposition with respect to the Cartan subalgebra 𝔥=𝑌2(1)𝑌2(2): 𝑌𝔥1(1)+𝑖𝑌3(1)𝑌1(2)+𝑖𝑌3(2)𝑌1(1)𝑌1(2).(4.2) Then 𝑌𝔪=(𝔥𝔪)1(1)𝑌1(2) with 𝑌𝔥𝔪=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 𝐽𝔫𝔫 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)𝑆𝜉33=01010000𝜉33,𝜉33,(5.1)(ii)𝐷𝜉11=𝜉11000𝜉110𝜉0011212𝜉11,𝜉11,𝜉110,(5.2)(iii)𝑇(𝑎,𝑏)=0𝑎𝑏01𝑏000𝑎𝑏1𝑏,𝑎,𝑏,𝑏0.(5.3) Any two distinct endomorphisms in the preceding list are nonequivalent. 𝑇(𝑎,𝑏) is equivalent to 𝑇(𝑎,𝑏)=𝑏𝑏0𝑎0000𝑎𝑏1𝑏.(5.4)

Proof. Let 𝐽=(𝜉𝑖𝑗)1𝑖,𝑗3 in the basis (𝑥1,𝑥2,𝑥3).The 9 torsion equations reduce to 𝜉13=𝜉23=0 and equation 123 (with the general notation introduced after Definition 2.6) which reads 𝜉33Tr(𝐴)=det(𝐴)1,(5.5) where 𝐴=𝜉11𝜉12𝜉21𝜉22. Suppose first Tr(𝐴)=0. Then 𝐴2=𝐼, so that 𝐴 is similar over , hence over , to 0110. Hence 𝐽010100𝜉33. Now, since 𝜉33 does not belong to the spectrum of 0110, taking the automorphism 100010𝛼𝛽1 of 𝔫 for suitable 𝛼,𝛽, one gets 𝐽𝑆(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)21)/2𝜉11𝐷(𝜉11). If 𝐴 is not a scalar matrix, then 𝐴 is similar to 0𝑎𝑏1𝑏 for some 𝑎,𝑏, and 𝑏0 from the trace. Then 𝐽𝑇(𝑎,𝑏). Finally, 𝑇(𝑎,𝑏)𝑇(𝑎,𝑏) since the matrices 0𝑎𝑏1𝑏 and 𝑏𝑏𝑎0 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 𝐽𝔫𝔫 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 01010000𝜉33,𝜉33.(6.1)
(ii) Any linear map 𝐽𝔫𝔫 which is an extension of a rank 1 𝐶𝑅-structure on 𝔫 such that 𝔭 is abelian is equivalent to a unique 𝜉1100001010,𝜉11.(6.2)𝐽 has nonzero torsion.

Proof. For any nonzero 𝑋𝔭, (𝑋,𝐽𝔭𝑋) is a basis of 𝔭. In case (i), [𝑋,𝐽𝔭𝑋]0, since 𝔭 is nonabelian. Then [𝑋,𝐽𝔭𝑋]=𝜇𝑥3, 𝜇0, and 𝑥3𝔭 since otherwise 𝔭 would be abelian. One may extend 𝐽𝔭 to 𝔫 in the basis (𝑋,𝐽𝔭𝑋,𝜇𝑥3) as 𝐽=01𝜉1310𝜉2300𝜉33,(6.3) and 𝐽 has zero torsion only if 𝜉13=𝜉23=0. In case (ii), necessarily 𝑥3𝔭 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 𝑦10 of 𝐽 associated to some eigenvalue 𝜉11. Then 𝑦1𝔭, which implies [𝑦1,𝐽𝑥3]0, for otherwise 𝑦1 would commute to the whole of 𝔫 and then be some multiple of 𝑥3𝔭. Hence [𝑦1,𝐽𝑥3]=𝜆𝑥3, 𝜆0, and dividing 𝑦1 by 𝜆 one may suppose 𝜆=1. In the basis 𝑦1,𝑦2=𝐽𝑥3,𝑦3=𝑥3 one has 𝜉𝐽=1100001010,(6.4) and (ii) follows.

Remark 6.2. Let 𝑁={[𝑥,𝑦,𝑧];𝑥,𝑦,𝑧} denote the Heisenberg group, where [𝑥,𝑦,𝑧]=1𝑥𝑧01𝑦001. 𝑁 can also be realized ([11, 12]) as the boundary 𝑀2={(𝜁,𝑤)2;𝜁=𝑤𝑤} of the Siegel half-space equipped with the multiplication (𝜁1,𝑤1)(𝜁2,𝑤2)=(𝜁1+𝜁2+2𝑖𝑤1𝑤2,𝑤1+𝑤2). The map Ψ𝑁𝑀2 defined by Ψ([𝑥,𝑦,𝑧])=(𝑧(1/2)𝑥𝑦+𝑖(𝑥2+𝑦2)/4,(1/2)(𝑥𝑖𝑦)) 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 2. 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 2, 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 𝑏Φ=11𝑏12𝑏000021𝑏22000000𝑏33𝑏340000𝑏43𝑏4400𝑏51𝑏52𝑏53𝑏54𝑏11𝑏22𝑏12𝑏210𝑏61𝑏62𝑏63𝑏640𝑏33𝑏44𝑏34𝑏43,𝑏11𝑏22𝑏12𝑏21𝑏33𝑏44𝑏34𝑏430.(7.1) The second kind ones are Ψ=ΘΦ where Φ is first kind and Θ=001000000100100000010000000001000010.(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)𝑆𝜀𝜉55=0100001000000001000010000000𝜉55𝜉𝜀552+10000𝜀𝜉55,𝜀=±1,𝜉55,(7.3)𝑆𝜀(𝜉55) is equivalent to 𝑆𝜀(𝜉55) (𝜀,𝜀=±1;𝜉55,𝜉55) under some first (resp., second) kind automorphism if and only if 𝜀=𝜀,𝜉55=𝜉55 (resp.,  𝜀=𝜀,𝜉55=𝜉55 ).(ii)𝐷𝜉11=𝜉11𝜉0112+10000𝜉11𝜉0112+10010𝜉11000010𝜉11𝜉00000011212𝜉11𝜉112+122𝜉11100002𝜉11𝜉11122𝜉11,𝜉11{0},(7.4)𝐷(𝜉11) is equivalent to 𝐷(𝜉11) (𝜉11,𝜉11) under some first (resp., second) kind automorphism if and only if 𝜉11=𝜉11 (resp., 𝜉11=𝜉11 ),(iii)𝑇𝜉33,𝜉43=0𝜉43𝜉33𝜉43𝜉33𝜉43𝜉331001𝜉33𝜉332+1𝜉43𝜉33𝜉33𝜉33000𝜉33𝜉33𝜉330010𝜉43𝜉000000043𝜉331𝜉33𝜉43𝜉33𝜉243𝜉33+𝜉332+1𝜉332𝜉0000143𝜉331𝜉33,𝜉33{0},𝜉43,(7.5)𝑇(𝜉33,𝜉43) is equivalent to 𝑇(𝜉33,𝜉43) (𝜉33,𝜉33{0},𝜉43,𝜉43) under some first (resp., second) kind automorphism if and only if 𝜉33=𝜉33,𝜉43=𝜉43 (resp.  𝜉33=𝜉33,𝜉43=𝜉43).
Finally, the cases (i), (ii), (iii) are mutually nonequivalent, either under first or second kind automorphism.

Proof. Let 𝐽=(𝜉𝑖𝑗)1𝑖,𝑗6 an integrable complex structure in the basis (𝑥𝑘)1𝑘6. Denote 𝐽1=𝜉11𝜉12𝜉21𝜉22, 𝐽2=𝜉13𝜉14𝜉23𝜉24, 𝐽3=𝜉31𝜉32𝜉41𝜉42, 𝐽4=𝜉33𝜉34𝜉43𝜉44. Then 𝐽1=𝜉11𝜉12𝜉15𝜉21𝜉22𝜉25𝜉51𝜉52𝜉55 and 𝐽3=𝜉33𝜉34𝜉36𝜉43𝜉44𝜉46𝜉63𝜉64𝜉66 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=01010000𝜉55, 𝐽3=01010000𝜉66,Case 2. 𝐽1=𝐷(𝜉11),𝐽3=𝐷(𝜉33), (𝜉11,𝜉330), andCase 3. 𝐽1=0𝜉1201𝜉22000𝜉55, 𝐽3=𝜉33𝜉330𝜉430000𝜉66, (𝜉22,𝜉330).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)Θ1. It is equivalent under some first kind automorphism to some 𝑇(𝜂33,𝜂43). That implies that the matrices 𝜉33𝜉33𝜉430, 0𝜂43𝜂331𝜂33 are similar, which amounts to their having same trace and same determinant, that is, 𝜂33=𝜉33,𝜂43=𝜉43. As 𝑇(𝜉33,𝜉43) is equivalent to 𝑇(𝜉33,𝜉43) under some second kind automorphism if and only if it is equivalent to Θ𝑇(𝜉33,𝜉43)Θ1 under some first kind automorphism, the assertion about second kind equivalence in Case 3 follows.

Remark 7.2. In Case 3, had we used 𝐽3=0𝜉3401𝜉44000𝜉66, then we would have to separate further into 2 subcases: subcase 𝜉120, 𝑇𝜉12,𝜉22=0𝜉12𝜉22𝜉12𝜉12+1001𝜉22𝜉12+1𝜉12𝜉22000𝜉120𝜉12001𝜉221𝜉22𝜉00000012+1𝜉22𝜉222+𝜉12+12𝜉22𝜉12𝜉000012𝜉22𝜉12+1𝜉22,𝜉12𝜉220,(7.6) subcase 𝜉12=0, 𝑇𝜉22=0010001𝜉220100100000𝜉22𝜉222+11𝜉221000000𝜉221𝜉22𝜉0000222+1𝜉221𝜉22,𝜉220.(7.7)

Remark 7.3. 𝑆𝜀(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 𝐽1 and 𝐽3 may be of different types.

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Copyright © 2011 Louis Magnin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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