#### Abstract

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

#### 1. Introduction

Given a real Lie algebra , the determination up to equivalence of zero torsion linear maps from to plays an important role in the computation of complex structures on direct products involving [1]. The direct computation of those maps can be difficult for semisimple , so there is a point in exploring alternative ways, particularly their relation to -structures. For compact , maximal rank -structures have been classified up to equivalence in [2]. In the case of , all zero torsion linear maps are extensions of certain -structures (see [1]). One can then ask the natural question whether or not any zero torsion linear map on a nonabelian is necessarily an extension of some -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: in the positive case, and 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 .

An interesting direction for future research could be to investigate zero torsion linear maps and -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 -Structures

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

Throughout this section, will denote any finite-dimensional real Lie algebra, its complexification, and or simply ^{β} the conjugation in with respect to .

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

*Definition 2.1. *A rank -structure on is a -dimensional complex subalgebra of such that .

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

Now one has the following straightforward lemma.

Lemma 2.2. *Let be vector subspace of and a linear map. Consider the real vector subspace of , where for . Denote . Then *(i)* is a complex vector subspace of if and only if ;*(ii)* is stable with respect to the bracket if and only if for any
In that case, with .*(iii)* is stable with respect to the bracket if and only if for any **
In that case, with for .*

*Remark 2.3. * When (i) is satisfied, (ii) and (iii) are trivially equivalent since then .

Lemma 2.4. *A rank -structure on can be defined in an alternative way as where is a -dimensional vector subspace of and is a linear map satisfying the 3 conditions
*

* Proof. *Let be a rank -structure on . Note first that the taking of the real part is a bijective linear map of the real algebra onto its image ,ββ, and there exists a unique linear map such that . Now, for ,ββ hence , so that . 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 and Trace , hence if , (2.4) and (2.5) follow from (2.3) and can be omitted.

*Definition 2.6. *A linear map is said to have zero torsion if it satisfies the condition

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

When computing the matrices of the zero torsion maps in some fixed basis of , we will denote by () the torsion equation obtained by projecting on the equation (2.6) with , .

The automorphism group ββ of acts on the set of all zero torsion linear maps and on the set of all complex structures on by for all ββ. It acts also on the set of -structures by . Two (resp., , ) are said to be *equivalent* (notation: (resp., )) if they are on the same ββ orbit. This means that the corresponding left invariant -structures on the connected simply connected real Lie group associated to are intertwined by some Lie group automorphism. It is a stronger notion than -diffeomorphy, where the intertwining is simply required to be a diffeomorphism.

Lemma 2.7. *Let be a linear map, and ; *(i)*,
*(ii)* is a real subalgebra of if and only if has zero torsion,*(iii)*if has zero torsion, is a complex subalgebra of .*

* Proof. *(i) For any one has

(ii) The result follows from Lemma 2.2(iii) since the first condition in (2.2) (with and ) is trivially satisfied and the second condition is the zero torsion condition.

(iii) From (ii), , hence and . Clearly is stable by multiplication by .

*Definition 2.8. *Let be a zero torsion linear map. We say that is an extension of a -structure if there exists a vector subspace of such that equipped with the restriction of to is a -structure on .

*Definition 2.9. *A real form of is said to be of type I (with respect to ) if . β is said to be type I if any real form of is of type I.

*Remark 2.10. * Introduce the real linear projections , defined by for . Then a real form of is of type I if and only if .

Proposition 2.11. *Let be a zero torsion linear map, and . β is an extension of a -structure if and only if .*

* Proof. *From Lemma 2.7, is a complex subalgebra of and . If is an extension of a -structure, one has hence . Conversely, if , let . Thenβ is stable under , and if denotes the restriction of to , conditions (2.3), (2.5) are trivially satisfied. Condition (2.4) holds true since, from Lemma 2.7(iii), is a subalgebra of . Precisely, for ,ββ hence and (2.4) follows.

Corollary 2.12. *There is a one-to-one correspondence between nontype I real forms of and zero torsion linear maps which are no extension of a -structure.*

* Proof. *Let be a zero torsion linear map that is no extension of a -structure. Then , hence is a real form of which is nontype I. Conversely, if is a nontype I real form of , then implies for some linear map , that is, . As is a real subalgebra, has zero torsion from Lemma 2.7(ii). Now since is a real form hence is not an extension of a -structure.

Corollary 2.13. *If is of type I, then any zero torsion linear map is an extension of a -structure.*

Proposition 2.14. *Let be any real form of ,ββ,ββ the conjugations with respect to ,ββ, and . If has a nonzero fixed point, then is type I.*

* Proof. *Let be a fixed point of . reads . Consider . Then . Hence has and similarly . Hence . Now, if and only if , that is, .

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

* Proof. *If is nonsolvable, so is . 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 of is type I.

#### 3. Case of

denotes the Lie group of real matrices with determinant 1 Its Lie algebra consists of the zero trace real matrices with basis , , and commutation relations Beside the basis , , , we will also make use of the basis where ,ββ,ββ, with commutation relations The adjoint representation of on is given by . The matrix of ( as in (3.1)) in the basis is The adjoint group is the identity component of ββ and one has The adjoint action of on preserves the form . The orbits are as follows: (i)the trivial orbit ;(ii)the upper sheet of the cone (orbit of ); (iii)the lower sheet of the cone (orbit of ); (iv)for all the one-sheet hyperboloid (orbit of ); (v)for all the upper sheet of the hyperboloid (orbit of )); (vi)for all the lower sheet of the hyperboloid (orbit of )).

The orbits of under the whole are as follows, beside : (I)the cone (orbit of ); (II)the one-sheet hyperboloid (orbit of ) (); (III) the two-sheet hyperboloid (orbit of )) ().

Proposition 3.1. *Let , and any linear map. has zero torsion if and only if it is equivalent to the endomorphism defined in the basis resp., by
** for (resp.,
** for ). *

* Proof. *Let in the basis . The 9 torsion equations are in the basis :
has at least one real eigenvalue . Let ,ββ, an eigenvector associated to . From the classification of the orbits of , we then get 3 cases according to whether is on the orbit (I), (II), (III)ββ(in the cases (II), (III) one may choose so that ). *Case 1. *There exists such that . Then, replacing by , we may suppose . That case is impossible from and .*Case 2. *There exists such that . Then we may suppose . Then from , , and , yield . Then and successively give and . Now and read respectively , and . Hence that case is impossible.*Case 3. *There exists such that . Then we may suppose that . Now instead of the basis , we consider the basis . The matrix of in the basis has the form
Then the 9 torsion equations (the star is to underline that the new basis is in use) for in that basis are as follows:
From and , *Case 1. *Suppose first that . Then .*Subcase 1. *Consider the subcase . and read, respectively, . Suppose . Then , and gives , which implies by . As then reads , this case is not possible. Now, the case is not possible either since then would read . We conclude that the Subcase 1 is not possible. *Subcase 2. *Hence we are in the Subcase 2:. Then . From , . Then yields and reads . This Subcase 2 is not possible either.*Case 2. *Hence Case 1 is not possible, and we are necessarily in the Case 2:. From (3.12), . Then reads hence and . From ,. Then reads , that is, , which implies . Now reads . The subcase is not possible since then would yield and would read . Hence . Then reads . The condition now implies the vanishing of all the torsion equations. In that case
Then in the basis
The cases are equivalent under .

*Remark 3.2. * is an extension of a -structure, in agreement with Corollary 2.15.

#### 4. Complex Structures on

We consider the basis of , with the upper index referring to the first or second factor. The automorphisms of fall into 2 kinds: the first kind is comprised of the , , and the second kind is comprised of the , with the switch between the two factors of .

Proposition 4.1. *Any integrable complex structure on is equivalent under some first kind automorphism to the endomorphism given in the basis by the matrix
** is equivalent to under some first (resp., second) kind automorphism if and only if ,β (resp., ,ββ).*

* Proof. *Let , ( blocks), an integrable complex structure in the basis . From Proposition 3.1, with some first kind automorphism, one may suppose ,ββ. As ,ββ. Then one is led to (4.1), and the result follows.

*Remark 4.2. * The complex subalgebra associated to has basis , , . The complexification of has weight spaces decomposition with respect to the Cartan subalgebra :
Then with , 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

Let be the real 3-dimensional Heisenberg Lie algebra with basis and commutation relations .

Proposition 5.1. *Let any linear map. has zero torsion if and only if it is equivalent to one of the endomorphisms defined in the basis by the following: **
Any two distinct endomorphisms in the preceding list are nonequivalent. is equivalent to
*

* Proof. *Let in the basis .The 9 torsion equations reduce to and equation (with the general notation introduced after Definition 2.6) which reads
where . Suppose first . Then , so that is similar over , hence over , to . Hence . Now, since does not belong to the spectrum of , taking the automorphism of for suitable , one gets . Suppose now . Then . If is a scalar matrix, that is, , then . If is not a scalar matrix, then is similar to for some , and from the trace. Then . Finally, since the matrices and are similar for they have the same spectrum and are no scalar matrices.

*Remark 5.2. * is an extension of a rank 1 -structure; however, are not.

#### 6. -Structures on

Proposition 6.1. *
(i) Any linear map which has zero torsion and is an extension of a rank 1 -structure on such that is nonabelian is equivalent to a unique
**
(ii) Any linear map which is an extension of a rank 1 -structure on such that is abelian is equivalent to a unique
** has nonzero torsion.*

* Proof. *For any nonzero , is a basis of . In case (i), , since is nonabelian. Then , , and since otherwise would be abelian. One may extend to in the basis as
and has zero torsion only if . In case (ii), necessarily since is abelian. Hence is a basis for . Take any linear extension of to . There exists some eigenvector of associated to some eigenvalue . Then , which implies , for otherwise would commute to the whole of and then be some multiple of . Hence , , and dividing by one may suppose . In the basis one has
and (ii) follows.

*Remark 6.2. * Let denote the Heisenberg group, where . can also be realized ([11, 12]) as the boundary of the Siegel half-space equipped with the multiplication . The map defined by is an isomorphism. If , denote the left invariant vector fields associated, respectively, to , then , hence the left invariant -structure on associated to the -structure on introduced in (i) is the -structure on induced from . The left invariant -structure on associated to the -structure on introduced in (ii) is not -diffeomorphic to the -structure on induced from , since the former has zero Levi form.

#### 7. Complex Structures on

We will use for commutation relations . The automorphisms of fall into 2 kinds. The first kind is comprised of the matrices The second kind ones are where is first kind and

Proposition 7.1. *Any integrable complex structure on is equivalent under some first kind automorphism to one of the following: ** is equivalent to () under some first (resp., second) kind automorphism if and only if (resp.,ββ ).** is equivalent to () under some first (resp., second) kind automorphism if and only if (resp., ),** is equivalent to () under some first (resp., second) kind automorphism if and only if (resp. β). **Finally, the cases (i), (ii), (iii) are mutually nonequivalent, either under first or second kind automorphism.*

* Proof. *Let an integrable complex structure in the basis . Denote , , , . Then and are zero torsion linear maps from to , 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 . Hence, modulo equivalence under some first kind automorphism, we get 3 cases: *Case 1. *, ,*Case 2. *, (), and*Case 3. *, , ().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 . It is equivalent under some first kind automorphism to some . That implies that the matrices , are similar, which amounts to their having same trace and same determinant, that is, . As is equivalent to under some second kind automorphism if and only if it is equivalent to under some first kind automorphism, the assertion about second kind equivalence in Case 3 follows.

*Remark 7.2. * In Case 3, had we used , then we would have to separate further into 2 subcases: subcase ,
subcase ,

*Remark 7.3. * is abelian (i.e., the corresponding complex subalgebra is abelian).

*Remark 7.4. * If one looks for zero torsion linear maps instead of complex structures, then and may be of different types.