On Complex Lie Algebroids with Constant Real Rank
We associate a real distribution to any complex Lie algebroid that we call the distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie al...
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| Дата: | 2025 |
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Інститут математики НАН України
2025
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| citation_txt | On Complex Lie Algebroids with Constant Real Rank. Dan Aguero. SIGMA 21 (2025), 044, 25 pages |
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| description | We associate a real distribution to any complex Lie algebroid that we call the distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles.
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 044, 25 pages
On Complex Lie Algebroids with Constant Real Rank
Dan AGUERO
Scuola Internazionale Superiore di Studi Avanzati - SISSA,
Via Bonomea, 265, 34136 Trieste, Italy
E-mail: dagueroc@sissa.it
Received September 30, 2024, in final form June 02, 2025; Published online June 13, 2025
https://doi.org/10.3842/SIGMA.2025.044
Abstract. We associate a real distribution to any complex Lie algebroid that we call
distribution of real elements and a new invariant that we call real rank, given by the pointwise
rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid
inside the original complex Lie algebroid. Under another regularity condition, we associate
a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide
a local splitting for complex Lie algebroids with constant real rank. In the last part, we
introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie
algebroid structures on the complexification of a vector bundle. We use this operation to
characterize all the complex Lie algebroid structures on the complexification of real vector
bundles.
Key words: complex Lie algebroids; Poisson geometry; normal forms
2020 Mathematics Subject Classification: 53D20
1 Introduction
Lie algebroids were introduced by Pradines [20], and nowadays have become a useful tool for
many areas of differential geometry and mathematical physics. This influenced their study and
made a fast development of the area. However, the study of Lie algebroids can be divided
into real, holomorphic, algebraic and complex Lie algebroids, with real Lie algebroids the most
studied. Complex Lie algebroids gained more attention after the introduction of generalized
complex structures [12, 13] and now we know that many geometrical structures are particular
examples of complex Lie algebroids, for example, generalized complex structures, complex Dirac
structures [2], exceptional complex structures [22], Bn-generalized complex structures [21], etc.
The theory of real and holomorphic Lie algebroids is currently more developed than the
theory of complex Lie algebroids. It is known their local structure, the obstructions for its
integrability (Lie III) [7, 16], etc. Despite the general theory of generalized complex structures
and complex Dirac structures being known [2, 3], it is still missing a general description of
complex Lie algebroids including its local description. About the integrability of complex Lie
algebroids there is not much known, some ideas can be found in [24].
In this article, we study complex Lie algebroids, by introducing a real distribution that we
call distribution of real elements and an invariant that we call the real rank which is just the
rank of the aforementioned distribution. When the real rank is constant the distribution of
real element is a real Lie algebroid and so we can associate a real foliation to the complex Lie
algebroid. In the philosophy of associating real geometrical structures, the Lie algebroid of real
elements is an analog to the Poisson and Dirac structures associated to generalized complex
and complex Dirac structures, respectively. We associate two other invariants: the type and
the class; and introduce the minimal complex Lie subalgebroids and the family of minimal
complex Lie algebroids. Our main examples are involutive structures, complexifications and
complex Poisson bivectors. We provide a local splitting theorem for complex Lie algebroids
mailto:dagueroc@sissa.it
https://doi.org/10.3842/SIGMA.2025.044
2 D. Aguero
with constant real rank. We also study complex Lie algebroid structures on complexifications of
real vector bundles, we introduce the complex matched pairs and an operation associated with
them that we call complex sum. We prove that there is a one-to-one correspondence between
complex Lie algebroid structures on complexifications and complex matched pairs.
The content of this article is organized as follows. In Section 2, we recall some properties of the
symmetries of complex vector bundles and complex anchored vector bundles. Since the author
has not found the proofs of some of the statements, they are provided. After that, we focus
on complex anchored vector bundles (CAVBs). We introduce the distribution of real elements
and its rank which we call real rank. When the distribution of real elements has constant rank
it is a real anchored vector bundle, in fact, it is involutive in case the CAVB is involutive.
In Section 3, we study complex Lie algebroids (CLAs), under the assumption of constant real
rank. We introduce the Lie algebroids of real elements denoted by (Are, [·, ·]re, ρre) and we study
some of its properties. This real Lie algebroid allows us to associate a real foliation to a CLA
with constant real rank. Since we are dealing with complex distributions, we can have many
types of regularity rather than the usual regularity (the image of the anchor map is a regular
distribution), so we introduce strongly regular CLAs. We also study the distribution given
by Amin = Are + iAre and whenever Amin is regular it is automatically a CLA, which we call
the minimal complex Lie subalgebroid. We introduce a new class of CLA that we call minimal
CLA, which are CLAs that coincide with their minimal complex subalgebroid. We also introduce
an N-valued function called the class that takes a point and returns the dimension of the leaf
passing through this specific point. CLAs of class 0 at a given point play an important role in
the splitting theorem for CLAS. In the last part of the section, we study the conjugate of a CLA.
In Section 4, we study the local structure of CAVBs and CLAs with constant real rank using
the techniques of [5]. We provide a proof of the splitting theorem for involutive CAVBs with
constant real rank and we end the section with the local description of CLAs with constant real
rank, obtaining the following:
Theorem 4.7. Let (A, [·, ·], ρ) be a CLA, m ∈M and ∆ = ρ(A)∩TM . Assume that (A, [·, ·], ρ)
has constant real rank in a neighborhood of m. Consider a submanifold N
ι
↪−→M such that
∆m ⊕ TmN = TmM and denote by P = ∆m. Then, there exists a neighborhood U of m such
that A|U is isomorphic to A′ × TCP , where (A′, [·, ·]′, ρ′) is a CLA of class 0 at m.
The property of A′ having class 0 at m already appears in the local description of generalized
complex structures and complex Dirac structures.
As a main application, we recover the Frobenius–Nirenberg theorem [19, Theorem 1].
In Section 5, we introduce the complex matched pairs of skew-algebroids. Given two skew-
algebroids (A, [·, ·]1, ρ1) and (A, [·, ·]2, ρ2) over the same vector bundle A→M , we say that they
form a complex matched pair if the brackets are compatible and the Jacobiator of the brackets
is the same. We observe that these pairs produce a complex Lie algebroid structure on the
complexification of a real vector bundle which we call complex sum. As an application, we prove
the following characterization:
Theorem 5.6. Let A be a real vector bundle over M . Any structure of CLA on AC comes from
a complex sum of a complex matched pair of skew-algebroids1 over A.
We also study the CLA associated to complex vector fields and describe its distribution of real
elements. In this case, we see that the assumption of constant real rank is inefficient for the local
description of general complex vector fields. However, allowing a “singular associated real Lie
algebroid” could provide a solution to this problem, this is the subject of future work. In the end
1Briefly speaking, a skew-algebroid is a Lie algebroid with a bracket that does not necessarily satisfy the Jacobi
identity, see Appendix A.
On Complex Lie Algebroids with Constant Real Rank 3
of the section, we study the CLA associated to a complex Poisson bivector from the perspective
of the complex sum. In the end of this article, we present two appendices: Appendix A, where
we recall some results and properties of real anchored vector bundles, vertical lifts and real Lie
algebroids, and Appendix B, where we recall some results and properties about Euler-like vector
fields and normal vector bundles.
Notation and convention
In this article, we only work with smooth manifolds. Given a vector bundle A over a smooth
manifoldM , a distribution is an assignmentm ∈M 7→ Rm ⊆ Am, where Rm is a vector subspace
of the fiber Am; when any v ∈ Rm can be extended to a local section of A taking values in R, we
say that the distribution is smooth. Each distribution has associated a N-valued function given
by the assignment rank: m ∈ M 7→ dimRm, which is called rank. A distribution is said to be
regular if it is a vector bundle.
Given a distribution E ⊂ AC, where AC is the complexification of a real vector bundle
A → M , we denote the space of real elements of E by ReE := E ∩ A and call them the real
part of E. Given a smooth map φ : M → N , we denote the complexification of Tφ by TCφ.
Throughout this article, a real anchored vector bundle is referred to by RAVB and a real Lie
algebroid by RLA. In Appendix A, we recall some properties of RAVBs and RLAs.
Let N be a submanifold of M , the normal bundle of N is denoted by
ν(M,N) = TM |N/TN,
when the context is clear we denote it by νN . In Appendix B, we recall some properties of
normal vector bundles.
2 Complex anchored vector bundles
2.1 Complex anchored vector bundles
A complex vector bundle E → M is equipped with a canonical bundle map jE ∈ Diff(E)
defined by jE(e) = i · e, the multiplication by the imaginary number i. The group of (complex)
automorphisms of E is
Aut(E) = {(Φ, φ) ∈ Aut(ER) | Φ ◦ jE = jE ◦ Φ and Φ covers φ ∈ Diff(M)},
where ER denotes the realification of E, and Aut(ER) is the automorphism group of ER as
a real vector bundle. The map jE acts on Aut(E) in the natural way. Consider a 1-parameter
subgroup {Φt}t∈R of Aut(E). Since Φt ∈ Aut(ER) and Φt ◦ jE = jE ◦ Φt, the vector field
X̂ = d
dtΦt|t=0 ∈ aut(ER) satisfies (jE)∗X̂ = X̂. So, the Lie algebra of (complex) infinitesimal
automorphisms of E is given by
aut(E) =
{
X̂ ∈ aut(ER) | (jE)∗X̂ = X̂
}
⊆ aut(ER) ⊆ X(E).
Another characterization of aut(E) is the following.
Lemma 2.1. The Lie algebra aut(E) is given by the pairs (L,X) such that X ∈ X(M) and
L : Γ(E) → Γ(E) is a C-linear derivation with respect to X.
Proof. Since X̃ ∈ aut(E) ⊆ aut(ER), it has associated a derivation given by the pair (L,X),
where L : Γ(ER) → Γ(ER) is R-linear and X = pr∗ X̃ ∈ X(M), defined as L(σ)↑ = [σ↑, X̃],
4 D. Aguero
∀σ ∈ Γ(E), where ↑ denotes the vertical lift, see Appendix A. Note that L is C-linear if and only
if L(iσ) = iL(σ) and thus
L(iσ)↑ =
[
(iσ)↑, X̃
]
=
[
(jE)∗σ
↑, X̃
]
=
[
(jE)∗σ
↑, (jE)∗X̃
]
= (jE)∗
[
σ↑, X̃
]
= (jE)∗
(
D(σ)↑
)
= (iL(σ))↑.
And by injectivity of the “↑”-map, we have that L(iσ) = iL(σ) and the lemma holds. ■
Definition 2.2. A complex anchored vector bundle (CAVB) is a pair (A, ρ), where A is a complex
vector bundle over M , and ρ : A → TCM is a C-linear bundle map. Let (A, ρA) and (B, ρB)
be two CAVBs over M and N , respectively. A morphism of complex anchored vector bundles
consists of a bundle map Φ: A → B and a map φ : M → N such that the following diagram
commutes:
A
ρA
��
Φ // B
ρB
��
TCM
TCφ // TCN.
Given a CAVB (A, ρ), the anchor map decomposes ρ = ρ1 + iρ2. So we obtain two real
anchored vector bundles (AR, ρ1) and (AR, ρ2). By the C-linearity of ρ, we have that
ρ1 ◦ jA = −ρ2.
Proposition 2.3. The map jA is an isomorphism of RAVBs between (A, ρ1) and (A,−ρ2).
The group of automorphisms of a CAVB (A, ρ) is given by
Aut(A, ρ) = {(Φ, φ) ∈ Aut(A) | ρ ◦ Φ = TCφ ◦ ρ}.
Note that ρ ◦ Φ = TCφ ◦ ρ if and only if ρ1 ◦ Φ = Tφ ◦ ρ1 and ρ2 ◦ Φ = Tφ ◦ ρ2. So
Aut(A, ρ) = Aut(AR, ρ1) ∩Aut(AR, ρ2) ∩Aut(A).
The (complex) infinitesimal automorphisms of (A, ρ) are denoted by aut(A, ρ).
Definition 2.4. The complex tangent extension of a vector field X ∈ X(M), is the vector field
XTC ∈ X(TCM) defined by the one-parameter subgroup TCφ
X
t , where φX
t is the local flow of X.
For a CAVB (A, ρ), we have that
aut(A, ρ) = aut(AR, ρ1) ∩ aut(AR, ρ2) ∩ aut(A)
=
{
X̃ ∈ X(A) | (jE)∗X̂ = X̂, X̃ ∼ρ XTC , where X̃ ∼pr X
}
.
In terms of the differential operators: a pair (L,X), where L : Γ(A) → Γ(A) is a C-linear
differential operator and X ∈ X(M), belongs to aut(A, ρ) if and only if ρ(L(τ)) = [X, ρ(τ)] or
equivalently
ρi(L(τ)) = [X, ρi(τ)] for i = 1, 2.
Definition 2.5. A real or complex anchored vector bundle (A, ρ) is involutive if ρ(Γ(A)) is a Lie
subalgebra of Γ(TM) or Γ(TCM), respectively.
Consider the decomposition of the anchor map ρ in its real and imaginary parts ρ = ρ1 + iρ2.
Note that the involutivity of (A, ρ) is not directly related to the involutivity of (AR, ρ1) and
(AR, ρ2). Even if both (AR, ρ1) and (AR, ρ2) are involutive, we cannot ensure the involutivity
of (A, ρ) and vice versa. To solve this issue, we shall introduce the distribution of real elements
in the next subsection.
On Complex Lie Algebroids with Constant Real Rank 5
2.2 Distributions associated to CAVBs
A CAVB (A, ρ) has associated two real distributions in TM :
∆ = Re
(
ρ(A) ∩ ρ(A)
)
and D = Re
(
ρ(A) + ρ(A)
)
.
Observe that D is always a smooth distribution, while ∆ is not necessarily smooth. However,
we have the following.
Lemma 2.6. If D is a regular distribution, then ∆ is a smooth distribution.
Proof. Consider the map ρ2 : A → TM , which is a real bundle map. Since the image of ρ2
is D, its kernel ker ρ2 is a vector bundle. Thus, ∆ = ρ(ker ρ2) is smooth. ■
Definition 2.7. Given a CAVB (A, ρ) define the distribution of real elements of (A, ρ) as
Are = {e ∈ A | ρ2(e) = 0},
which we called distribution of real elements of (A, ρ). In general, Are is a distribution in A (not
necessarily smooth). When Are is regular, the pair (Are, ρre) is a RAVB, where ρre = ρ1|Are .
The N-valued function
real-rankA : m ∈M 7→ dimRA
re|m
is called real rank of A.
Constant real rank is equivalent to Are being a vector bundle and by the proof of Lemma 2.6,
it is equivalent to the regularity of D as a distribution. Indeed, we can see that
real-rankA+ rankRD = rankRAR. (2.1)
Lemma 2.8. Assume that (A, ρ) has constant real rank. If (L,X) ∈ aut(A, ρ), then L(Γ(Are)) ⊆
Γ(Are) and (L|Are , X) ∈ aut(Are, ρre).
Proof. Straightforward. ■
Lemma 2.9. If (A, ρ) is an involutive CAVB with constant real rank, then (Are, ρre) is involutive.
Proof. Let α, β ∈ Γ(Are), equivalently, ρ(α) and ρ(β) are real, and so [ρ(α), ρ(β)]. By in-
volutivity of A, there exists γ ∈ Γ(A) such that [ρ(α), ρ(β)] = ρ(γ). So ρ(γ) is real and
consequently γ ∈ Γ(Are). ■
The image of the anchor map of an involutive RAVB is integrable, see for example [5].
In the case of a CAVB (A, ρ), it follows from the proof of Lemma 2.6 and by Lemma 2.9 that
the real part of ρ(A) is integrable.
The pullback of complex anchored vector bundle (A, ρ) over M along the map φ : N →M is
defined in the same way as the real case, as the fibered product:
φ!A
��
// A
ρ
��
TCN
TCφ // TCM.
Note that
φ!A =
{
(a,X1 + iX2) | (a,X1) ∈ φ!A1 and (a,X2) ∈ φ!A2
}
= φ!A1 ×prA iφ!A2 ⊆ A× TCN,
where A1 = (AR, ρ1) and A2 = (AR, ρ2), and prA : A × TCN → A is the obvious projection.
As a consequence, we can see that
6 D. Aguero
1. φ!TCM = TCN .
2. If N =M ×Q, then pr!M A = A× TCQ.
3. If N
ι
↪−→M is a submanifold transversal to ρ, then N is also transversal to ρ1 and ρ2.
Moreover,(
ι!A
)
R = ρ−1(TCN)R = {a ∈ AR | ρ(a) = ρ1(a) + iρ2(a) ∈ TCN} = ι!A1 ∩ ι!A2.
4. If ρ is injective, then φ!A = (TCφ)
−1(A).
The following lemma is straightforward.
Lemma 2.10. Let (A, ρ) be a CAVB and a map φ : N →M . Then,(
φ!A
)re
= φ!(Are)×prA i(A× kerTφ),
∆φ!A = Im(Tφ) ∩∆A.
A submanifold N
ι
↪−→M is called transversal to the CAVB if
ρ(A) + TCN = TCM
(or the stronger condition ∆+ TN = TM). By transversality, ι!A is smooth and so is a CAVB.
Note that transversality with respect to ρ implies transversality with respect to A1 and A2.
A direct consequence of the Lemma 2.10 is that
(
ι!A
)re
= ι!(Are) for submanifolds N
ι
↪−→M
(not necessarily transversal to the anchor map). The following lemma appeared originally in [5],
and the proof is similar in the complex case.
Lemma 2.11. Suppose (A, ρ) is an involutive CAVB over M and φ : N →M a map transversal
to ρ. Then φ!A is an involutive CAVB.
3 Complex Lie algebroids
Definition 3.1. A complex Lie algebroid (CLA) over a manifoldM is a complex anchored vector
bundle (A, ρ) over M together with a Lie bracket on Γ(A) satisfying the Jacobi identity
[·, ·] : Γ(A)× Γ(A) → Γ(A)
and with a C-bundle map ρ : A → TCM , that we call the anchor map, preserving brackets and
satisfying the Leibniz identity
[α, fβ] = f [α, β] + ρ(α)(f)β
for all α, β ∈ Γ(A) and for any function f ∈ C∞(M,C). A complex skew-algebroid is when the
Jacobi identity is not necessarily satisfied, and an almost complex Lie algebroid (almost CLA)
is a complex skew-algebroid where the anchor preserves the brackets.
Remark 3.2. Observe that CLAs are defined on the category of smooth real manifolds and not
on the category of holomorphic manifolds.
Morphisms between CLAs are defined in the exact same way as RLAs, and so are their
automorphisms.
On Complex Lie Algebroids with Constant Real Rank 7
Examples 3.3. Some examples of CLAs are the following:
1. An involutive structure E is an involutive vector subbundle of TCM , see [23]. The triple
(E, [·, ·], ρ), where [·, ·] is the complexification of the Lie bracket of vector fields and ρ is the
inclusion map of E in TCM , defines a CLA. In particular: complex structures, transverse
holomorphic structures and CR structures define CLAs.
We recall that a CR structure is a pair (I,H), where H is a subbundle of TM and
I : H → H is a bundle map satisfying that I2 = −Id and the following: if X,Y ∈ H, then
[IX, Y ] + [X, IY ] ∈ H and I([IX, Y ] + [X, IY ]) = [IX, IY ]− [X,Y ].
In this case, the triple (E, [·, ·], ι) is a CLA, where E = ker(IC−iId) ⊆ TCM , the bracket [·, ·]
is the complexification of the Lie bracket of vector fields and ι : E → TCM is the inclusion
map. The case of complex and transverse holomorphic structures is similar.
2. Complex Dirac structures [2], generalized complex structures [11], Bn-generalized complex
structures [21] and exceptional generalized complex structures [22]. Let L be a complex
Dirac structure, that is, a lagrangian subbundle L of (TM ⊕T ∗M)C which is closed under
the Courant–Dorfman bracket J·, ·K. The Courant–Dorfman bracket when restricted to
the sections of an involutive isotropic subbundle become a Lie bracket. Thus, the CLA
structure on L is given by the triple (L, J·, ·K|Γ(L)×Γ(L), prTCM ). It is similar for the other
aforementioned structures.
3. A holomorphic Lie algebroid (A, [·, ·], ρ) has associated two CLAs, (A, [·, ·]1,0, ρ1,0) and
(A, [·, ·]0,1, ρ0,1) defined in [15, Section 3.4].
4. A bundle of complex Lie algebras is a complex vector bundle p : E → M where each
fiber E|m is equipped with a complex Lie algebra bracket [·, ·]m and satisfies the following:
if s1, s2 ∈ Γ(E), then the assignment
m ∈M → [s1, s2]|m := [s1|m, s2|m]m ∈ E|m
is a smooth section of E. Then, the triple defined by (E, [·, ·], ρ = 0) is a CLA. Conversely,
any CLA (A, [·, ·], ρ) having ρ = 0 is a bundle of complex Lie algebras (the proof is almost
the same as in the real case, see for example [6, Section 16.2]).
5. The complexification (AC, [·, ·]C, ρC) of a RLA (A, [·, ·], ρ) defines a CLA.
6. Given a complex Lie algebra g we call a Lie algebra homomorphism X : g → Γ(TCM)
a complex infinitesimal action. The vector bundle L(g) = M × g is a CLA with anchor
map X and bracket
[u, v](m) = [u(m), v(m)] +
(
LX (u(m))v
)
(m)−
(
LX (v(m))u
)
(m),
where u, v and [u, v] are sections of L(g) taken as maps from M to g.
7. A complex Poisson bivector is a bivector π ∈ Γ
(
∧2T ∗
CM
)
satisfying [π, π] = 0. The triple(
T ∗
CM, [·, ·]π, π
)
is a CLA, where the bracket [·, ·]π is defined as
[α, β]π = Lπ(α)β − Lπ(β)α− dπ(α, β)
and anchor π : T ∗
CM → TCM . We denote this complex Lie algebroid by
(
T ∗
CM
)
π
. This is
the complex parallel of the Lie algebroid associated to a Poisson structure.
8. A complex vector field Z = X1 + iX2 ∈ Γ(TCM) defines a structure of CLA over the
bundle M × C that we denote by AZ . The anchor map is given by
ρ : Γ(M × C) = C∞(M,C) → TCM, ρ(f) = fZ
and the bracket by
[f, g] = fLZ(g)− gLZ(f).
8 D. Aguero
3.1 Associated RLA
On one hand, a RLA have associated a distribution given by the image of the anchor map.
On the other hand, the image of the anchor map of a CLA is a complex distribution of TCM .
To bypass this issue, we associate two real distributions:
∆ = Re
(
ρ(A) ∩ ρ(A)
)
and D = Re
(
ρ(A) + ρ(A)
)
. (3.1)
Note that D is a smooth distribution, while ∆ is not smooth in general, see [2, Example 6.1].
Given a CLA (A, [·, ·], ρ), decompose ρ = ρ1 + iρ2. Contrary to what happens with CAVBs,
neither A1 = (AR, [·, ·], ρ1) nor A2 = (AR, [·, ·], ρ2) necessarily define RLAs, since ρ1 and ρ2 do
not preserve the bracket (and so they do not satisfy the Leibniz identity).
Actually, the natural candidate for A2 is (AR, i[·, ·],−ρ2), here we write −ρ2 in order to
avoid iρ2. The failure of the Leibniz identity for A1 and A2 is controlled in the following way:
[a, fb]1 − (f [a, b]1 + (ρ1(a)f)b) = i(ρ2(a)f)b, (3.2)
[a, fb]2 − (f [a, b]2 − (ρ2(a)f)b) = i(ρ1(a)f)b, (3.3)
where [·, ·]1 = [·, ·] and [·, ·]2 = i[·, ·].
An immediate consequence of equations (3.2) and (3.3) is the following.
Proposition 3.4. Let (A, [·, ·], ρ) be a CLA. If either A1 = (AR, [·, ·], ρ1) or A2 = (AR, i[·, ·],−ρ2)
are RLAs, then ρ = 0.
In general, A1 and A2 satisfy the Jacobi identity but they are not RLAs, since there is an
“error term” in the Leibniz identity controlled by the anchor of the other algebroid.
We avoid these error terms by working with ker ρ1 and ker ρ2. Also note that the multiplica-
tion by i is a kind of “isomorphism” from A1 to A2.
In the same way as with CAVBs, we consider the distribution
Are = ker ρ2 = ρ−1(∆)
and the real rank of a CLA as
real-rankA : m ∈M 7→ dimRA
re|m.
We also have the distribution given by ker ρ1. However, note that
ker ρ1 = jA(A
re) = ρ−1(i∆),
where jA is the multiplication by i in the fibers. Moreover,
[Are, Are] ⊆ Are, [Are, jA(A
re)] ⊆ jA(A
re) and [jA(A
re), jA(A
re)] ⊆ Are. (3.4)
Consider
ρre = ρ1|Are and ρim = −ρ2|jA(Are).
As a consequence, we get the following.
Proposition 3.5. If (A, [·, ·], ρ) is a CLA with constant real rank, then (Are, [·, ·]|Are , ρ
re) and
(jA(A
re), jA[·, ·]|jA(Are)
, ρim) are isomorphic RLAs, with the isomorphism given by the restric-
tion of jA. Moreover, ρre(Are) = ρim(jA(A
re)) = ∆.
On Complex Lie Algebroids with Constant Real Rank 9
Definition 3.6. When Are is a RLA, we shall call it as the Lie algebroid of real elements.
One of the key feature of the Lie algebroid of real elements is that it allows us to associate
a foliation to any CLA with constant real index.
Corollary 3.7. If a CLA (A, [·, ·], ρ) has constant real index, then the set of real elements of the
image of ρ, Re ρ(A) coincides with the image of ρ(Are). Consequently, Re ρ(A) is an integrable
distribution.
A RLA is said to be regular when the image of the anchor map is a subbundle of TM . CLAs
have three associated distributions: the image of the anchor map, ∆ and D, see equation (3.1).
Therefore, a regularity condition for CLAs should take these two additional distributions into
consideration.
Definition 3.8. A CLA (A, [·, ·], ρ) is regular if ρ(A) is a regular distribution and we say that
it is strongly regular if D and ∆ are regular distributions.
Note that strongly regular CLAs are regular CLAs with constant real rank, and vice versa.
Remark 3.9. Another natural invariant associated to a CLA is the order, which is a N-function
defined as follows:
orderA : m ∈M 7→ corankRD|m.
The order was previously introduced in [2] for complex Dirac structures and in [22], under the
name “class”, for exceptional complex structures. By equation (2.1), we have that
orderA|m = real-rankA|m + dimM − rankAR, (3.5)
and so, constant order is equivalent to constant real rank.
In general, the distribution D is not integrable, even for strongly regular CLAs. Non-Levi-
flat CR structures provide examples of strongly regular CLAs where the distribution D is not
integrable.
Definition 3.10. Let (A, [·, ·], ρ) be a CLA (or an RLA) and letm ∈M be a point. The isotropy
Lie algebra of A at m is the complex (or real) Lie algebra
gm(A) = ker ρm.
When A is regular, we obtain a bundle of complex (or real) Lie algebras given by g(A) = ker ρ.
Note that
g(A)R = Are ∩ jA(Are) and g(Are) = g(jA(A
re)) = g(A)R.
3.2 Minimal complex subalgebroid and the class
Given a CLA (A, [·, ·], ρ), we associate the following smooth real distribution:
Amin = Are + jA(A
re).
Since Amin is jA-invariant, it is a complex distribution. In what follows in this article Amin
will be considered as a complex distribution. Denote by [·, ·]min = [·, ·]|Amin
and ρmin = ρ|Amin .
An immediate consequence of equation (3.4) is the following.
Proposition 3.11. If Amin is a regular distribution, then the triple (Amin, [·, ·]min, ρmin) is a com-
plex Lie subalgebroid of A, satisfying (Amin)
re = Are.
10 D. Aguero
Assuming, for example, that A is strongly regular, Amin is a vector bundle. One immediate
consequence of the previous proposition is that Amin is the minimal complex Lie subalgebroid
of A with the same associated RLA as A.
Definition 3.12. We call Amin the minimal complex subalgebroid of A. We say that A is
a minimal CLA whenever A = Amin.
Given a strongly regular CLA (A, [·, ·], ρ), we canonically associate to it two CLAs: Amin
and (Are)C. These are related by the following canonical exact sequence of CLAs:
0 // g(A)
Φ // (Are)C
Ψ // Amin
// 0, (3.6)
where the maps Φ and Ψ are Lie algebroid morphisms given by
Φ: g(A) → (Are)C, a 7→ a− ijAa,
Ψ: (Are)C → Amin, a+ ib 7→ a+ jAb.
As a consequence of the exact sequence (3.6), we obtain the following.
Corollary 3.13. Let A be a CLA with constant real rank such that Amin is regular. Then,
Amin
∼= (Are)C if and only if A is an involutive structure (see Examples 3.3).
Note that Amin = ρ−1(∆C). Since A is strongly regular, g(A) is a bundle of complex Lie
algebras, and we have the following exact sequence:
0 // g(A) // Amin
// ∆C // 0.
Now we extend the notion of type of complex Dirac structures [2, Definition 4.7] to CLAs.
Definition 3.14. The type of A at the point m ∈M is defined as
typeA|m =
1
2
(dimD|m − dim∆|m).
Note that CLAs with type 0 are precisely those whose anchor map has a real image ρ(A)=DC,
where D is a real distribution. Also note that the type takes values between 0 and dimM
2
or dimM−1
2 , depending on the parity of dimM . The type has a different meaning in the more
general context of CLAs.
Proposition 3.15. The following identity holds:
typeA|m = rankCA− rankCAmin|m.
Proof. Applying the rank-nullity theorem on ρ and ρmin, we have
rankC ρ(A) + rankC g(A) = rankCA and rankC∆C + rankC g(A) = rankCAmin.
Subtracting the previous expression, we obtain the desired identity. ■
Corollary 3.16. A CLA has type 0 if and only if it is minimal.
Thus, the type measures how close is a CLA to being minimal. Furthermore, a strongly
regular CLA has constant type, and ρ(A) defines a transverse CR structure on M (see [1,
Definition 2.74]).
We introduce the following invariant.
On Complex Lie Algebroids with Constant Real Rank 11
Definition 3.17. The class of a CLA A at the point m ∈M is given by
classA|m = rank∆|m.
We say that a CLA is of class k if it has constant class k.
Proposition 3.18. We have the following identities:
classA|m = rankCAmin|m − rankC g(A)|m = rankRA
re|m − rankR g(A)R|m.
While the type measures how close a CLA is to being minimal, the class measures how close
it is to being “real-degenerate”, i.e., when Are = g(A)R. Note that the class is bounded below
by 0 but it is not bounded above. We shall see in Examples 3.20 and 3.22 that complexifications
provide examples of CLAs with arbitrarily high class. A CLA has class 0 at a point m ∈ M if
and only if ∆|m = 0. CR structures and holomorphic Lie algebroid are examples of CLAs of
class 0, as we shall see in Examples 3.20 and 3.22. It is easy to see that the only constant real
rank CLAs with type 0 and class 0 are bundles of complex Lie algebras.
The real rank, the type and the class are related by the following.
Proposition 3.19. The following identity holds:
2 type+ class+ real-rank = rankRAR.
Proof. Applying the rank-nullity theorem on ρ and ρ2 we have the following:
2 type = 2(rankRD − rankC ρ(A))
= 2(rankRAR − rankRA
re)− 2(rankCA− rankC g(A))
= rankRAR − 2 real-rank+ rankR g(A)R
= rankRAR − class− real-rank . ■
Example 3.20 (involutive structures). Given an involutive structure (E, ρ, [·, ·]), we have the
following:
Ere = ∆ = E ∩ TM, Emin = ∆C.
Moreover,
real-rankE = rank∆, typeE = rankE − rank∆ and classE = rank∆.
Note that CR structures have trivial real distribution. In fact, they are the only CLAs with
this property. Indeed, assume that Are = 0, then A is an involutive because g(A) = 0 and
Are = A ∩ TM = 0.
Proposition 3.21. Let A be a CLA. Then, Are = 0 if and only if A is a CR structure.
Examples 3.22.
1. Complex Dirac structures: In the case of a complex Dirac structure L ⊆ TCM , see [2], we
have
Lre = L ∩
(
TM ⊕ T ∗
CM
)
.
Any complex Dirac structure L with constant order has associated a real Dirac structure L̂.
By the proof of [2, Theorem 5.1], we have the following exact sequence of Lie algebroids:
0 // L ∩ T ∗M // Lre Φ // L̂ // 0,
12 D. Aguero
where
Φ: Lre → L̂, X + iξ + η 7→ X + ξ.
In particular, if L is a generalized complex structure, then Lre ∼= L̂. By equation (3.5),
we have
real-rankL = orderL+ dimM.
In general, complex Dirac structures are not minimal algebroids. By Corollary 3.16, only
complex Dirac structures with type 0 are minimal.
2. Holomorphic Lie algebroids: A holomorphic Lie algebroid (A, [·, ·], ρ) is seen as a CLA via
its associated CLA (A, [·, ·]1,0, ρ1,0), see [15]. Note that ρ1,0(A) ⊆ T0,1M and so ∆ = 0.
Hence, holomorphic Lie algebroids have class 0
3. Bundle of complex Lie algebras: In this case, the anchor map is zero, so ∆ = 0. Moreover,
Are = AR and Amin = A.
4. Complexification of Lie algebroids: Let (A, [·, ·], ρ) be a RLA and let (AC, [·, ·]C, ρC) be its
complexification. Note that ∆ = D = ρ(A) and so AC is minimal. Furthermore,
(AC)
re = A⊕ i ker ρ.
As a consequence,
real-rankAC = rankRA+ rankR g(A), classAC = rankRA− rankR g(A).
In particular, A has constant real rank if and only if A is a regular RLA.
5. Complex Poisson bivectors: A complex Poisson bivector π ∈ Γ
(
∧2TCM
)
decomposes as
π = π1 + iπ2, where π1 and π2 are real bivectors (not necessarily Poisson). Denote the
anchor map of
(
T ∗
CM
)
π
by ρ. Then, ρ is expressed in terms of π1 and π2 in the following
way:
ρ(ξ + iη) = π(ξ + iη) = π1(ξ)− π2(η) + i(π2(ξ) + π1(η)),
where ξ, η ∈ T ∗M . So, the associated RLA is given by((
T ∗
CM
)
π
)re
= {ξ + iη | π2(ξ) + π1(η) = 0}
with anchor map ρre(ξ + iη) = π1(ξ) − π2(η). So we can associate a real foliation to
a complex Poisson bivector, which is not necessarily a symplectic foliation. The RLA(((
T ∗
CM
)
π
)re
, [·, ·]re, ρre
)
and its associated foliation are subject of future work.
Consider two CLAs (A1, [·, ·]1, ρ) and (A2, [·, ·]2, ρ̃) over the same manifold M , and a Lie
algebroid morphism ψ : A1 → A2. Decompose the anchor maps into their real and imaginary
components ρ = ρ1 + iρ2 and ρ̃ = ρ̃1 + iρ̃2. Note that ψ ◦ jA1 = jA2 ◦ ψ, and ρ1 = ρ̃1 ◦ ψ and
ρ2 = ρ̃2 ◦ ψ. As a consequence of these relations, we have the following.
Proposition 3.23. A morphism ψ between two CLAs A1 and A2, satisfies the following:
ψ((A1)
re) ⊆ (A2)
re and ψ((A1)min) ⊆ (A2)min.
In particular, ψ|(A1)re and ψ|(A1)min
are Lie algebroid morphisms in both cases. Moreover, both
the real rank and the class are invariants under Lie algebroid automorphisms.
On Complex Lie Algebroids with Constant Real Rank 13
Remark 3.24. Making a parallel with complex Dirac structures: given a complex Dirac struc-
ture L ⊆ TCM with constant real index, we associate to it two Lie algebroids: the RLA
K = Re
(
L ∩ L
)
and the CLA KC. Note that we are strongly using the fact that L is
a subbundle of TCM , where a natural complex conjugation is available. Although, in general,
a CLA (A, [·, ·], ρ) is not naturally realized as a subbundle of the complexification of a Courant or
another Lie algebroid, the image of the anchor map ρ(A) is a distribution of TCM . Thus, we pass
the information of the real elements of ρ(A) via the inverse image of the anchor map. Therefore,
instead of K we consider Are = ρ−1(∆) and instead of KC we consider Amin = ρ−1(∆C).
3.3 Conjugate CLA
We recall that given a complex vector bundle A, its conjugate bundle A is the vector bundle
defined by the following modification on the rule of scalar multiplication:
z ·A e = z ·A e
for all z ∈ C and e ∈ A, see [17].
Let (A, ρ, [·, ·]) be a CLA. The map ρ becomes C-linear when defined on A. The bracket
[·, ·] : Γ
(
A
)
×Γ
(
A
)
→ Γ
(
A
)
is well-defined and remains C-bilinear. Then, we have the following.
Proposition 3.25. The triple
(
A, ρ, [·, ·]
)
is a CLA.
Proof. Since the bracket remains unchanged, the Jacobi identity is satisfied. It remains to
verify the Leibniz identity. For e1, e2 ∈ Γ
(
A
)
and f ∈ C∞(M,C), we have[
e1, f ·A e2
]
=
[
e1, f ·A e2
]
= f [e1, e2] +
(
ρ(e1)f
)
·A e2
= f [e1, e2] + ρ(e1)f ·A e2 = f [e1, e2] +
(
ρ(e1)f
)
·A e2. ■
We refer to the triple
(
A, ρ, [·, ·]
)
as the conjugate complex Lie algebroid.
4 Local structure
In this section, we study the local structure of CAVBs and CLAs with constant real rank.
Our main tools for providing these local structures come from the techniques developed in [5].
In Appendix B, we recall the properties of normal vector bundles and Euler-like vector fields,
which play a key role in this section.
4.1 Local description of involutive CAVBs
Involutive RAVBs inherit the structure of an almost Lie algebroid [5, Proposition 3.17]. For in-
volutive CAVBs, we have the same result and the proof is an adaptation of the real case to the
complex case.
Proposition 4.1. A real or complex anchored vector bundle (A, ρ) is involutive if and only if
there exists a bracket on Γ(A) making A into a real or complex almost Lie algebroid, respectively.
Let (A, ρ) be an involutive CAVB with constant real rank. Consider a bracket [·, ·] that
makes (A, ρ) into an almost Lie algebroid. Then, we have an operator Dσ : Γ(A) → Γ(A), given
by Dστ = [σ, τ ]. As mentioned in [4], we have the map
ρ̃ : Γ(Are) → aut(A, ρ), ρ̃(σ) = (Dσ, ρ(σ))
14 D. Aguero
that fits into the following diagram:
aut(A, ρ)
��
Γ(Are)
ρ̃
99
ρ // X(M).
(4.1)
Note that, the map ρ̃ lifts to a CAVB automorphism but not necessarily to a complex almost
Lie algebroid automorphism. We shall use the following identification of vector bundles.
Lemma 4.2. Let N
ι
↪−→M be a submanifold. Then, the vector bundles ν(TCM,TCN)
prTCN−−−−→TCN
and TCν(M,N)
TC prN−−−−→ TCN are isomorphic as real vector bundles.
Proof. We recall that TCM = TM×M TM and so T (TCM) = TTM×TM TTM . Consequently,
T (TCM)|TCN = (TTM ×TM TTM)|TN×NTN = TTM |TN ×TM |N TTM |TN .
Note that
ν(TCM,TCN) = (T (TCM)|TCN )/T (TCN)
=
(
TTM |TN ×TM |N TTM |TN
)
/TTN ×TN TTN
∼= ν(TM, TN)×ν(M,N) ν(TM, TN)
∼= TCν(M,N). ■
Remark 4.3. Using the identification of Lemma 4.2, we see that if N
ι
↪−→M is a transversal
to ρ = ρ1 + iρ2, then the anchor map is a map of pairs ρ :
(
A, ι!A
)
→ (TCM,TCN), and
ν(ρ) : ν
(
A, ι!A
)
→ ν(TCM,TCN) ∼= TCν(M,N) is given by
ν(ρ) = ν(ρ1) + iν(ρ2).
Lemma 4.4. If X is an Euler-like vector field along N (see Definition B.2), then its complex
tangent extension XTC (see Definition 2.4) is Euler-like along TCN .
Proof. It is easy to see that XTC |TCN = 0. Let E denote the Euler vector field associated
to ν(M,N) → N . Since the complexified tangent lift of the scalar multiplication is the scalar
multiplication in TCν(M,N) → TCN , we note that ETC is an Euler vector field. Using the
identification of Lemma 4.2, we see that ν(XTC) = ν(X)TC = ETC . ■
We now apply the techniques of [5] to the case of CAVBs.
Proposition 4.5. Let (A, ρ) be an involutive CAVB with constant real rank and let N
ι
↪−→M be
a submanifold transverse to ρre. Then
1. There exists X̃ ∈ aut(A, ρ) such that X̃|ι!A = 0 and X = (prTM )∗X̃ is Euler-like along N .
2. Each choice of vector field X̃ as in (1) determines an isomorphism of CAVBs
ψ̃ : p!ι!A→ A|U ,
whose base map is the tubular neighborhood embedding ψ : νN → U ⊆ M (see Defini-
tion B.2).
On Complex Lie Algebroids with Constant Real Rank 15
Proof. (1) The proof is similar to [5, Theorem 3.13], we just have to work on TCM instead
of TM .
(2) By Lemma B.3, consider ϵ ∈ Γ(Are) such that ρ(ϵ) = X is Euler-like along N and also
consider the lifting of equation (4.1), X̃ ∈ aut(A, ρ), given by X̃ = ρ̃(ϵ). Note that X̃ is ρ-related
to XTC , by the description of aut(A, ρ). Then ν
(
X̃
)
is ν(ρ)-related to ν(XTC) = ETC , the last one
is an Euler vector field and by Lemma B.1, ν(ρ) is a fibre-wise isomorphism. Consequently, X̃ is
an Euler-like vector field along ι!A. Consider Φt and Φ̃t the flow of X and X̃, respectively. Since
both are Euler-like, the maps λt = Φ− log(t) and λ̃t = Φ̃− log(t) are defined at t = 0. Note that
λt ◦ ψ = ψ ◦ κt, in the case of t = 0 this means that λ0 ◦ ψ = p ◦ ι, where ψ : νN → U ⊆ M
is the tubular neighborhood embedding associated to X. Recall that the maps Φ̃t are CLA
automorphisms with base Φt, so the maps λ̃t restricted to A|U are automorphism of CLA with
base λt, for all t ≥ 0. Since λ̃t ◦ λ̃s = λ̃t+s, we have that λ̃t ◦ λ̃t = λ̃2t and taking limit to t = 0,
we get that λ̃20 = λ̃0 and so λ̃0 is a projection. Given that X̃|ι!A = 0, we have that λ̃t|ι!A = Idι!A,
so taking limit λ̃0|ι!A = Idι!A, so ι
!A ⊆ Im
(
λ̃0
)
and using the fact that
ρ ◦ λ̃0 = TCλ ◦ ρ = TCp ◦ TCψ−1 ◦ ρ, (4.2)
we obtain that Im
(
λ̃0
)
= ι!A. Consider the map
Ψ: A|U → ι!A× TCνN , a ∈ A 7→
(
λ̃0(a), TCψ
−1(ρ(a))
)
.
By equation (4.2), the image of Ψ lies in p!ι!A. Finally, note that Ψ is an isomorphism from A|U
to p!ι!A, and we define ψ̃ = Ψ−1 as the desired map. ■
Corollary 4.6. Let (A, ρ) be an involutive CAVB and m ∈M . Assume that (A, ρ) has constant
real rank in a neighborhood of m. Let N
ι
↪−→M be a submanifold such that ∆|m⊕TmN = TmM ,
and set P = ∆|m. Then, there exists a neighborhood U of m such that A|U is isomorphic
to ι!A× TCP .
Proof. Since N is completely transversal to P = ∆|m , we can choose a trivialization of the
normal bundle νN = N × P . Then, by Proposition 4.5 (2), we have p!ι!A = ι!A× TCP . ■
4.2 Local description of CLA
Consider a CLA (A, [·, ·], ρ) with constant real rank. Since A is a CLA, there exists a natural
lift ρ̃ : Γ(Are) → autCLA(A) of the anchor map ρ, as in equation (4.1), now defined by the Lie
algebroid bracket.
Theorem 4.7. Let (A, [·, ·], ρ) be a CLA with constant real rank, and let N
ι
↪−→M be a subman-
ifold transverse to Are (and so transverse to A). Choose a section ϵ ∈ Γ(Are), such that ϵ|N = 0
and ρ(ϵ) ∈ X(M) is Euler-like along N . Then, there exists a tubular neighborhood embedding
ψ : νN → U ⊆M , depending on the choice of ϵ, and a Lie algebroid isomorphism
ψ̃ : p!ι!A→ A|U
covering the tubular neighborhood embedding ψ : νN → U ⊆M
Proof. The proof follows the same strategy as that of Proposition 4.5, now using the lift
described above. Thus, we obtain a map ψ̃ : p!ι!A −→ A|U that is an isomorphism as CAVBs, we
just have to check that ψ̃ preserves the Lie algebroid bracket. To this end, we use the argument
of [5, Remark 3.20]. Consider the family of maps
Ψt : A|U → ι!A× TCνN , a ∈ A 7→
(
λ̃t(a), TCψ
−1(ρ(a))
)
.
16 D. Aguero
Note that the image of Ψt is κ
!
tψ
!A and consider the maps ψ̂t = Ψ−1
t . Since each map ψ̂t preserves
the bracket for all t > 0, it follows that their limit ψ̂0 = ψ̃ also preserves it. This completes
the proof. ■
Corollary 4.8. Let (A, [·, ·], ρ) be a CLA and m ∈M . Assume that (A, [·, ·], ρ) has constant real
rank in a neighborhood of m. Consider a submanifold N
ι
↪−→M such that ∆m ⊕ TmN = TmM
and set P = ∆m. Then, there exists a neighborhood U of m such that A|U is isomorphic
to ι!A× TCP .
Note that ι!A has class 0 at m and TCP is a minimal CLA (see Examples 3.22 (4)), so the
previous description tells us that a CLA with constant real rank splits in a neighborhood of
a point m as a product of a minimal CLA with a CLA of class 0 at m. The splitting theorem for
complex Dirac structures states that a complex Dirac structure with constant order decomposes
around a point m ∈M as B-transformation of the product of a complex Dirac structure having
associated Poisson structure vanishing at m (a CLA having class 0 at m) with the complex
Dirac structure associated to the presymplectic leaf passing through m (a minimal CLA, see
Examples 3.22 (1)), see [2]. With this in mind, we can say that minimal CLAs and class 0 CLA
are fundamental pieces for the splitting.
Remark 4.9. If N
ι
↪−→M is a submanifold transversal to ∆, then a normal form ψ̃ : p!ι!A→ A|U
induces a normal form for Are. Indeed, the transversality of N with respect to ∆, implies the
transversality of N with respect to the anchor map ρ. By Lemma B.3, there exists ϵ ∈ Γ(Are)
such that ρ1(ϵ) = X is Euler-like along N , this is the same as in the proof of (2). By Lemma 2.8,
the restriction of the lift ρ̃ to Are produces a lifting ρ̂ for Are. So ρ̂(ϵ) = X̂ satisfies the hypotheses
of Theorem A.2. This determines an isomorphism of RAVBs, ψ̂ : p!ι!Are → Are|U with the
same base map ψ of Proposition 4.5.
Assuming the conditions of Corollary 4.8, suppose further that A has constant real rank and
constant type around a point m ∈ M . Then ∆ is a regular distribution and in a neighborhood
U ⊆ M of m, we have that TN |U∩N ⊕∆|U∩N = TM |U∩N and so ι!A is of class 0. Hence, we
obtain the following.
Corollary 4.10. If (A, [·, ·], ρ) is a strongly regular CLA, m ∈ M and P = ∆m, then in
a neighborhood of m, the bundle A splits as the product of a CLA of class 0 with TCP . Moreover,
if (A, [·, ·], ρ) is a minimal CLA with constant real rank, then around m, we have that A is
isomorphic to the product of a bundle of complex Lie algebras with TCP .
Example 4.11 (ivolutive structures). We apply the splitting Theorem 4.7 for CLAs to an
involutive structure E.
Proposition 4.12. Let (E, ρ, [·, ·]) be an involutive structure and assume that Ere = ∆ is
regular. Consider m ∈ M and a submanifold N
ι
↪−→M passing through m, which is completely
transversal to ∆, i.e., ∆m ⊕ TmN = TmM . Then, there exists a neighborhood U of m such that
E|U ∼= ι!E × TCP,
where P = ∆m and ι!E is a CR structure over U ∩N .
Proof. By Theorem 4.7, there exists a neighborhood U of m such that E|U ∼= ι!E × TCP .
Since ∆ is regular, shrinking U if necessary, we have that TN |U∩N ⊕ ∆|U∩N = TM |U∩N .
As a consequence, ι!E has no real elements, i.e., ι!E ∩ ι!E = 0 and so it is a CR structure. ■
As a consequence, we recover a classical result of Nirenberg about the local description of
involutive structures.
On Complex Lie Algebroids with Constant Real Rank 17
Corollary 4.13 ([19, Theorem 1]). Let (E, ρ, [·, ·]) be an involutive structure such that ∆ =
E∩TM and D =
(
E+E
)
∩TM are regular involutive distributions. Then, for any point m ∈M ,
there exists a neighborhood U of m and a coordinate system (xl, yl, pj , qk), with l = 1, . . . , e− r,
j = 1, . . . , r and k = 1, . . . , n− 2e+ r, where dimM = n, rankE = e and dim∆ = r, such that
E|U = SpanC
{
∂
∂zl
,
∂
∂pj
}
,
where zl = xl + iyl.
Proof. Choose a submanifoldN
ι
↪−→M satisfying that TmN⊕∆m = TmM . By Proposition 4.12,
there exists a neighborhood U of m such that E|U ∼= ι!E × TCP , where ι
!E is a CR structure.
Since D is involutive and N transversal to D, we have that D′ = Re
(
ι!E + ι!E
)
is an involutive
distribution on U ∩N . Consequently, ι!E is a Levi-flat CR structure. The generators ∂
∂zl
come
from the holomorphic foliation associated to ι!E, while the generators ∂
∂pj
come from TCP . ■
Example 4.14 (complex infinitesimal actions). Let g be a complex Lie algebra and χ : g →
Γ(TCM) a complex infinitesimal action. Decompose χ as χ = χ1+iχ2, where χ1, χ2 : g → Γ(TM)
are R-linear. Assume that L(g)re is a trivial vector bundle. Then, we have
L(g)re = ker ρ2 =M × gre,
where gre = kerχ2 is a real Lie algebra contained in g. Moreover, χ1|gre → Γ(TM) is an
infinitesimal action on M and so it defines a RLA, that we denote by L(gre). Let m ∈ M , and
let S be an orbit of L(gre) passing throughm. Choose a submanifoldN
ι
↪−→M passing throughm,
which is completely transversal to S. Then, by Corollary 4.8, there exists a neighborhood U
of m such that
L(g)|U ∼= ι!L(g)× TCS.
In case the map χ maps into complete vector fields, the infinitesimal action χ1|gre is complete
and so by Lie–Palais theorem, this infinitesimal action integrates to a Lie group action σ : Gre →
Diff(M), where Gre is the Lie group integrating gre. In this case, S is also the orbit of the action
on the point m.
Given a RLA (A, [·, ·], ρ), its complexification (AC, [·, ·]C, ρC) has constant real rank if and only
if A is a regular RLA; see Examples 3.22 (4). So Corollary 4.8 does not directly apply in this case.
However, the conclusions of Corollary 4.8 are still true for the complexification of any RLA and
this follows from the local structure of the original A. Indeed, consider a submanifold N
ι
↪−→M
transversal to ρC and a point m ∈M . Then, N is also transversal to ρ and by [5, Corollary 4.2],
there exists a neighborhood U of p and an isomorphism of Lie algebroids Ψ: A|U → ι!A× TCP ,
where P = ∆|m. Note that
(
ι!A
)
C = ι!(AC) and so, the complexification of Ψ
ΨC : (AC)|U → ι!(AC)× TCP
gives the desired local isomorphism. Summarizing, we have the following.
Proposition 4.15. Let (A, [·, ·], ρ) be a RLA and let m ∈M . Consider (AC, [·, ·]C, ρC) with the
usual CLA structure given by the complexification of A and let N
ι
↪−→ M be a submanifold such
that ∆m ⊕ TmN = TmM . Denote by P = ∆m. Then, there exists a neighborhood U of m such
that (AC)|U is isomorphic to ι!(AC)× TCP .
Remark 4.16. Applying Corollary 4.8 to complex Dirac structures with constant order, we ob-
tain a different result from the one presented in [2]. The reason is that the group of Lie algebroid
automorphisms is larger than the group of automorphisms of complex Dirac structures, so the
local description provided here is weaker than the one given in [2], in terms of automorphisms.
However, in [2] we assume that L has constant real index, a condition that is not assumed here.
18 D. Aguero
5 Complex sum of skew-algebroids
Let A be a real vector bundle. Given two brackets [·, ·]1 and [·, ·]2 on Γ(A), we say that they
commute or that they are compatible if the following identity holds:
([a, [b, c]1]2 + c.p.) + ([a, [b, c]2]1 + c.p.) = 0 (5.1)
for all a, b, c ∈ Γ(A).
Definition 5.1. Let A → M be a real vector bundle and A1 = (A, [·, ·]1, ρ1) and A2 =
(A, [·, ·]2, ρ2) be two structures of skew-algebroids on A. The complex sum of A1 and A2 is
the CAVB (AC, [·, ·], ρ), with bracket defined by
[a, b] = [a, b]1 + i[a, b]2
for a, b ∈ Γ(A) and then extended by C-linearity to Γ(AC), and with anchor map
ρ(a+ ib) = ρ1(a)− ρ2(b) + i(ρ1(b) + ρ2(a)).
Proposition 5.2. Let A1 and A2 be as above. Then
1. The complex sum always satisfies the Leibniz identity, and so it is a complex skew-algebroid.
2. The complex sum is an almost CLA (the anchor map preserves the brackets, see Ap-
pendix A) if and only if
ρ1[a, b]2 + ρ2[a, b]1 = [ρ2(a), ρ1(b)] + [ρ1(a), ρ2(b)].
3. The complex sum is a CLA if and only if the skew-algebroids commute and have the same
Jacobiators Jac[·, ·]1 = Jac[·, ·]2.
Proof. (1) and (2): It is enough to prove this for real elements.
(3): The commutativity of the brackets and the equality of the Jacobiator of [·, ·]1 and [·, ·]2
are together equivalent to the Jacobi identity for the bracket [·, ·]. ■
As a consequence of the previous proposition, we have the following.
Definition 5.3. A pair of skew-algebroids structures (A1,A2) over the same vector bundle A
is called a complex matched pair if they satisfy the hypothesis of Item (3) of Proposition 5.2.
Remark 5.4. There is a definition of matched pairs for real Lie algebroids, which can be
extended to CLAs. We shall refer to this extension as C-extended matched pairs. Our definition
is slightly different. While a C-extended matched pairs consists of a pair of CLAs (A,B) over the
same base manifoldM , such that C = A⊕B is a CLA overM andA and B are subalgebroids of C,
see [18, Definition 4.1], our definition could be seen as a pair (A1,A2) of skew-algebroids over
the same vector bundle A→M such that “A1+iA2” is a CLA, as defined above. Moreover, A1
and A2 are not necessarily subalgebroids.
Examples 5.5.
1. The complexification of a RLA (A, [·, ·], ρ) is the complex sum of (A, [·, ·], ρ) with
(A, [·, ·] = 0, 0).
2. The complex double of a RLA (A, [·, ·], ρ) is the complex sum of (A, [·, ·], ρ) with itself.
In this case
Are = {a+ i(γ − a) | a ∈ A, γ ∈ ker ρ}.
On Complex Lie Algebroids with Constant Real Rank 19
Theorem 5.6. Let A be a real vector bundle over M . Any structure of CLA on AC comes from
a complex sum of a complex matched pair of skew-algebroids.
Proof. Consider the R-linear maps ρ̃1, ρ̃2 : AC → TM given by the decomposition ρ = ρ̃1+iρ̃2.
Note that
ρ(a+ ib) = (ρ̃1(a)− ρ̃2(b)) + i(ρ̃1(b) + ρ̃2(a)).
Now consider the R-linear maps ρ1 = ρ̃1|A and ρ2 = ρ̃2|A. Then
ρ(a) = ρ1(a) + iρ2(a) ∀ a ∈ A.
Now consider the R-linear maps B1, B2 : Γ(AC)× Γ(AC) → Γ(A) defined as follow:
[a1 + ia2, b1 + ib2] = B1(a1 + ia2, b1 + ib2) + iB2(a1 + ia2, b1 + ib2).
In the same way, as with the decomposition of ρ, we get that
[a1, a2] = B1(a1, a2) + iB2(a1, a2) ∀ a1, a2 ∈ Γ(A). (5.2)
Consider the brackets [·, ·]1 = B1|Γ(A)×Γ(A) and [·, ·]2 = B2|Γ(A)×Γ(A) on Γ(A). The conditions
of item (3) of Proposition 5.2, follow from examining the real and imaginary parts appearing
in the Jacobi identity of [·, ·] after decomposing it in terms of B1 and B2. The real part yields
Jac[·, ·]1 = Jac[·, ·]2, while the imaginary part is equivalent to the compatibility condition for
the brackets.
Using equation (5.2) and the Leibniz identity, we have that
B1(a1, fa2) + iB2(a1, fa2) = [a1, fa2] = f [a1, a2] + (ρ(a1)f)a2
= fB1(a1, a2) + (ρ1(a1)f)a2 + i(fB2(a1, a2) + (ρ2(a1)f)a2),
so both [·, ·]1 and [·, ·]2 satisfy the Leibniz identity. The triples (A, [·, ·]1, ρ1) and (A, [·, ·]1, ρ1) do
not necessarily satisfy the Jacobi identity, so they are skew-algebroids and their complex sum
is (AC, [·, ·], ρ) by construction. ■
The distributions associated to a complex sum AC are the following:
D = (ρ1(A) + ρ2(A))C,
∆ = {ρ1(a)− ρ2(b) | ρ2(a) + ρ1(b) = 0 and a, b ∈ A} and
(AC)
re = {a+ ib ∈ AC | ρ2(a) + ρ1(b) = 0}.
Let A1 and A2 be two skew-algebroid structures on A. Then, they have associated fiber-wise
linear bivectors π1 and π2 on A∗, respectively (see Appendix A).
Proposition 5.7. The compatibility condition of the brackets is equivalent to the compatibility
condition of the bivectors [π1, π2] = 0. The condition Jac[·, ·]1 = Jac[·, ·]2 is equivalent to the
condition [π1, π1] = [π2, π2].
Proof. First, we recall the identity
[π1, π2](f, g, h) = π1(π2(f, g), h) + π2(π1(f, g), h) + c.p.,
where π1, π2 ∈ X2(M), f, g, h ∈ C∞(M,R), and we see the multivectors as multi-derivations.
The result follows by evaluating this identity on linear and basic functions of C∞(A∗,R). ■
These equivalences provide geometric interpretations of the conditions defining a complex
matched pair.
20 D. Aguero
5.1 Complex vector fields
As an application, we study the structure of CLAs associated to complex vector fields, see
Example 3.3 (8). The following Proposition is a well-known fact, but we provide a proof for
completeness:
Proposition 5.8. The triple (AZ , [·, ·], ρ) is a CLA.
Proof. It is straightforward to verify that the conditions for being an almost CLA are satis-
fied. For the second part, note that AZ , as a bundle, is the complexification of M × R and
that (AZ , [·, ·], ρ) is the complex sum of the Lie algebroids AX1 and AX2 , where Z = X1 + iX2,
X1, X2 ∈ X(M). So, by Proposition 5.6, AZ is a CLA if and only if the brackets of AX1
and AX2 commute, since the equality of the Jacobiators is already satisfied (both have vanishing
Jacobiators). The brackets of AX1 and AX2 commute if and only if equation (5.1) is satisfied
([f, [g, h]1]2 + c.p) + ([f, [g, h]2]1 + c.p.) = 0.
Expanding the terms, we have that
[f, [g, h]1]2 + [f, [g, h]2]1 = fg(LX1LX2h+ LX2LX1h)− fh(LX1LX2g + LX2LX1g)
− g(LX1hLX2f + LX2hLX1f) + h(LX1gLX2f + LX2gLX1f).
The remaining terms are permutations of this formula. Note that all these terms cancel each
other out and so the proposition holds. ■
Now we study the distribution (AZ)
re. First, note that
ρ|m(z1 + iz2) = z1X1|m − z2X2|m + i(z2X1|m + z1X2|m).
By analyzing the different cases, we calculate (AZ)
re = ker ρ2. In summary, we obtain the
following description:
(AZ)
re|m =
Cm, X1|m = X2|m = 0,
(R⊕ i0)|m, X1|m ̸= 0, X2|m = 0,
(0⊕ iR)|m, X1|m = 0, X2|m ̸= 0,
0m,
X1|m ̸= 0, X2|m ̸= 0,
X1|m, X2|m are linearly independent,
{−ct+ it | t ∈ R}m, X1|m ̸= 0, X2|m ̸= 0 and X2|m = cX1|m
and
(AZ)min|m =
{
0, X1|m ̸= 0, X2|m ̸= 0, X1|m, X2|m are linearly independent,
Cm, elsewhere.
Finally, we have the following.
Proposition 5.9. The distribution AZ has constant real rank if and only if either
1. X1 = X2 = 0.
2. X1|m ̸= 0, X2|m ̸= 0 and X1|m and X2|m are linearly independent.
3. X1 and X2 never vanish simultaneously, and there exist smooth functions c ∈ C∞(M \
Sing(X1)) and d ∈ C∞(M \ Sing(X2)) such that cX1 = X2 and dX2 = X1.
On Complex Lie Algebroids with Constant Real Rank 21
As a consequence of Proposition 5.9, in cases (1) and (2) the foliation associated to AZ
consists of isolated points, while in case (3) by the integral curves of X1 or X2, depending on
which one does not vanish.
Proposition 5.10. Any structure of CLA over M ×C →M comes from a complex vector field
and is a complex sum of the RLAs defined by the real and imaginary components of this complex
vector field.
Proof. Let (M ×C, [·, ·], ρ) be any structure of CLA over M ×C. Consider the complex vector
field Z given by Zm = ρ(m, 1) or equivalently Z = ρ(1), where 1 ∈ C∞(M,C) is the constant
function. So, for any f ∈ C∞(M,C), we have ρ(f) = fZ. By the Leibniz identity, we have
[1, f ] = [1, f · 1] = LZf
and thus
[f, g] = [f, g · 1] = g[f, 1] + (ρ(f)g)1 = fLZg − gLZf.
Consider X1, X2 ∈ X(M) such that Z = X1 + iX2. By evaluating the anchor map and the
bracket on real functions, it follows that
ρ(f) = fX1 + ifX2 and [f, g] = [f, g]1 + i[f, g]2,
where [·, ·]1 and [·, ·]2 are the brackets of the RLAs defined by X1 and X2. ■
5.2 Complex bivectors
Consider a real bivector γ ∈ Γ
(
∧2TM
)
and the bracket
[·, ·]γ : Γ(T ∗M)× Γ(T ∗M) → Γ(T ∗M)
defined by
[α, β]γ = Lγ(α)β − Lγ(β)α− dγ(α, β) (5.3)
for α, β ∈ Γ(T ∗M). For a complex bivector π ∈ Γ
(
∧2TCM
)
, the same formula as in (5.3) defines
a bracket [·, ·]π on Γ(TCM). A straightforward verification yields the following.
Proposition 5.11. If γ is a real or complex bivector, then the triple (T ∗M, [·, ·]γ , γ) or
(
T ∗
CM,
[·, ·]γ , γ
)
is a skew-algebroid or a complex skew-algebroid, respectively. We denote these structures
by (T ∗M)γ or
(
T ∗
CM
)
γ
, respectively.
Consider a complex bivector π = π1 + iπ2 ∈ Γ
(
∧2TCM
)
, where π1 and π2 are real bivectors.
Denote by ρ the anchor map of
(
T ∗
CM
)
π
. Then, ρ decomposes as
ρ(ξ + iη) = π(ξ + iη) = π1(ξ)− π2(η) + i(π2(ξ) + π1(η))
for ξ, η ∈ T ∗M . Moreover, the bracket evaluated on real elements satisfies
[α, β]π = [α, β]π1 + i[α, β]π2 .
Proposition 5.12. Let π = π1 + iπ2 be a complex bivector. Then, (T ∗
CM)π is the complex sum
of (T ∗M)π1 and (T ∗M)π2. Moreover, π is a complex Poisson bivector if and only if (T ∗M)π1
and (T ∗M)π2 form a complex matched pair.
22 D. Aguero
Proof. The first part of the proposition follows from what was exposed above. The bivector π
is Poisson if and only if
(
T ∗
CM
)
π
is a CLA. So by item (3) of Proposition 5.2, this holds if and
only if (T ∗M)π1 and (T ∗M)π2 form a complex matched pair. ■
Remark 5.13. In particular, we note that a pair of Poisson bivectors (π1, π2) is a bi-Hamiltonian
structure if and only if (T ∗M)π1 and (T ∗M)π2 form a complex matched pair. By combining
Propositions 5.7 and 5.12, we note that the notion of complex matched pair on a pair of Lie
algebroids (A1,A2) actually extends the notion of bi-Hamiltonian structures to the context of
Lie algebroids. This extension leads to the notion of “bi-Hamiltonian Lie algebroids”.
A Anchored vector bundles and Lie algebroids
In this appendix, we recall some properties of real anchored vector bundles and real Lie alge-
broids.
Definition A.1. A real anchored vector bundle (RAVB) is a pair (A, ρ), where A is a vector
bundle over a manifold M and ρ : A → TM is a R-bundle morphism called the anchor map.
We say that a RAVB is involutive whenever ρ(Γ(A)) is a Lie subalgebra of Γ(TM).
Given two RAVBs (A, ρA) and (B, ρB) over M and N , respectively. A morphism of RAVBs
is given by a bundle map Φ: A → B and a map φ : M → N such that the following diagram
commute:
A
ρA
��
Φ // B
ρB
��
TM
Tφ // TN.
We recall that the Lie algebra aut(E) of infinitesimal automorphism of a real vector bundle E
are vector fields X̃ ∈ X(E) whose flow is given by automorphism of E. Equivalently, they
are given by derivations, that is, pairs (L,X), where L : Γ(E) → Γ(E) is a R-linear operator
and X ∈ X(M) satisfying L(fσ) = fL(σ) +X(f)σ. The relationship is given by the following:
given X̃ ∈ aut(E), consider the pair (L,X) defined by
X = pr∗ X̃, pr∗⟨η,X⟩ =
〈
pr∗ η, X̃
〉
and L(σ)↑ =
[
σ↑, X̃
]
,
where σ ∈ Γ(E), η ∈ Ω1(M) and σ↑ is the vertical lift of σ:
σ ∈ Γ(E) → σ↑ ∈ X(E), σ↑(em) =
d
dt
∣∣∣∣
t=0
(em + tσ(m)) ∀ em ∈ Em.
Note that this map is injective, since σ↑ = 0, implies that σ = 0 just by evaluating in the zero
section of E.
The space of automorphisms of a RAVB (A, ρ) is given by
Aut(A, ρ) = {(Φ, φ) ∈ Aut(A) | ρ ◦ Φ = Tφ ◦ ρ}
and the space of infinitesimal automorphisms of a RAVB (A, ρ) is given by
aut(A, ρ) =
{
X̃ ∈ aut(A) | X̃ ∼ρ XT
}
,
where XT ∈ X(TM) is the vector field constructed with the differential of the flow of X. Equiv-
alently, the infinitesimal automorphisms of (A, ρ) are given by derivations (L,X) satisfying that
ρ(L(σ)) = [X, ρ(σ)].
On Complex Lie Algebroids with Constant Real Rank 23
The pullback of a RAVB (A, ρ) over M along the map φ : N → M is defined as the fibered
product:
φ!A
��
// A
ρ
��
TN
Tφ // TM.
The local description of an involutive RAVB is the following.
Theorem A.2 ([5, Corollary 3.14]). Let (A, ρ) be an involutive RAVB, m ∈ M , and N
ι
↪−→M
a submanifold containing m satisfying that ρ(A|m) ⊕ TmN = TmM . Let P = ρ(A|m). Then,
there exists a neighborhood U of m such that A|U is isomorphic to ι!A× TP .
Definition A.3. A real Lie algebroid (RLA) over a manifold M is a triple (A, [·, ·], ρ), where
(A, ρ) is a RAVB over M together with a Lie bracket on Γ(A):
[·, ·] : Γ(A)× Γ(A) → Γ(A)
satisfying the Leibniz property
[α, fβ] = f [α, β] + ρ(α)(f)β
for all α, β ∈ Γ(A) and f ∈ C∞(M). If the triple (A, [·, ·], ρ) satisfies all the previous condi-
tions except the Jacobi identity, it is called a skew-algebroid, see [10, Definition 4.1]; they were
originally introduced in [14] as differential pre-Lie algebras.
Skew-algebroids A are equivalent to fiber-wise linear bivectors on A∗, in the following way:
{f ◦ pA∗ , g ◦ pA∗} = 0, {lα, f ◦ pA∗} = ρ(α)f, {lα, lβ} = l[α,β],
where pA∗ : A∗ → M is the usual bundle projection, f, g ∈ C∞(M,R), α, β ∈ Γ(A) and
lα, lβ ∈ C∞(A∗,R) are defined as lα(τ) = ⟨τ, α⟩. In case these bivectors are Poisson bivectors,
we obtain Lie algebroid structures.
The pullback of RLAs is defined in the same way as the pullback of RAVBs. We recall the
local description of RLAs.
Theorem A.4 ([5, 9, 8]). Let (A, [·, ·], ρ) be a RLA and m ∈ M . Consider a submanifold
N
ι
↪−→M such that ρ(A|m) ⊕ TmN = TmM and denote by P = ρ(A|m). Then, there exists
a neighborhood U of m such that A|U is isomorphic to ι!A× TP .
The following table summarizes all the algebroids used in this article:
Real Complex
Real anchored vector bundle (RAVB): Complex anchored vector bundle (CAVB):
R-vector bundle and R-anchor map C-vector bundle and C-anchor map
Involutive RAVB: Involutive CAVB:
A RAVB where the image of the anchor map A CAVB where the image of the anchor map
is involutive in Γ(TM) is involutive in Γ(TCM)
Real skew-algebroid: Complex skew-algebroid:
R-vector bundle, R-anchor map and C-vector bundle, C-anchor map and
R-bilinear bracket satisfying only Leibniz C-bilinear bracket satisfying only Leibniz
Real almost Lie algebroid: Complex almost Lie algebroid
It is a real skew-algebroid where It is a complex skew-algebroid where
the anchor map preserves the bracket the anchor map preserves the bracket
Real Lie algebroid (RLA): Complex Lie algebroid (CLA):
It is a real skew-algebroid where It is a complex skew-algebroid where
the bracket satisfies Jacobi the bracket satisfies Jacobi
24 D. Aguero
B Euler-like vector fields and normal forms
In this appendix we recall the properties of normal vector bundles and Euler-like vector fields,
that are mainly used in Section 4. The Euler vector field associated to a vector bundle E →M ,
usually denoted by E ∈ X(E), is the vector field generated by the one-parameter group s 7→ κe−s ,
where the maps κt : E → E are given by κte = t · e, t ∈ R (in case t = 0, κ0 is the projection to
the zero section).
Given a submanifold N of M , we denote the normal bundle of N by
ν(M,N) = TM |N/TN,
when the context is clear we denote ν(M,N) by νN and the projection map by p : ν(M,N) → N .
Any map of pairs φ : (M ′, N ′) → (M,N) has an associated map ν(φ) : ν(M ′, N ′) → ν(M,N).
The following lemma is a known fact.
Lemma B.1. Let φ : M ′ → M be a smooth map. If N
ι
↪−→M is a submanifold transverse to φ
and N ′ = φ−1(N), then ν(φ) is a fiberwise isomorphism.
Consider a vector field X ∈ X(M) tangent to N , that is X : (M,N) → (TM, TN). Using the
fact that TνN = νTN (see [5, Appendix A]), we have that ν(X) is a vector field on ν(M,N).
Definition B.2. Let N be a submanifold of M . A tubular neighborhood embedding is an em-
bedding ψ : νN →M that takes N to N and ν(ψ) = Id, where ν(ψ) is induced by ψ : (νN , N) →
(M,N) (here we are using the identification ν(νN , N) = νN ). A vector field X ∈ X(M) is called
Euler-like (along N) if it is complete, X|N = 0 and ν(X) is the Euler vector field of ν(M,N).
Given a submanifold N of M , there is a one-to-one correspondence between Euler-like vector
and tubular neighborhood embedding, see, for example, [5].
Lemma B.3 ([5, Lemma 3.9]). Let (A, ρ) be RAVB over M , and N ⊆ M a submanifolds
transversal to ρ. Then there exists a section ϵ ∈ Γ(A) with ϵ|N = 0, such that X = ρ(ϵ) is
Euler-like along N .
Given a vector bundle E
p−→M , we know that
TE
Tp−→ TM and TE ×E TE
Tp×Tp−−−−→ TM ×M TM
are vector bundles. We give the structure of complex vector bundle to the last one and we call
it the complexified tangent bundle TCE
TCp−−→ TCM , with the fiber sum and fiber multiplication
by scalars given by
(V + iW ) +TCp
(
V ′ + iW ′) = (
V +Tp V
′)+ i
(
W +Tp W
′),
(λ1 + iλ2) ·TCp (V + iW ) = (λ1 ·Tp V − λ2 ·Tp W ) + i(λ2 ·Tp V + λ1 ·Tp W ),
where V , V ′, W and W ′ ∈ TCE, with p(V ) = p(V ′), p(W ) = p(W ′) and λ1, λ2 ∈ R, and +Tp
and ·Tp denotes the operations of fiber sum and multiplication by scalar in the tangent vec-
tor bundle TE
Tp−→ TM . Note that the previous operations make TCE
TCp−−→ TCM a complex
vector bundle,
Γ
(
∧kT ∗
CM
)
Γ
(
∧kT ∗M
)
C
Γ
(
∧k+1T ∗
CM
)
Γ
(
∧k+1T ∗M
)
C.
d̃
̂
dĈ
On Complex Lie Algebroids with Constant Real Rank 25
Acknowledgements
The author acknowledges Henrique Bursztyn for proposing the topic and his many valuable
observations. The author is also grateful to Hudson Lima, Roberto Rubio and Pedro Frejlich
for many fruitful and helpful conversations. We are also thankful to the anonymous referees for
their valuable comments and suggestions.
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1 Introduction
2 Complex anchored vector bundles
2.1 Complex anchored vector bundles
2.2 Distributions associated to CAVBs
3 Complex Lie algebroids
3.1 Associated RLA
3.2 Minimal complex subalgebroid and the class
3.3 Conjugate CLA
4 Local structure
4.1 Local description of involutive CAVBs
4.2 Local description of CLA
5 Complex sum of skew-algebroids
5.1 Complex vector fields
5.2 Complex bivectors
A Anchored vector bundles and Lie algebroids
B Euler-like vector fields and normal forms
References
|
| id | nasplib_isofts_kiev_ua-123456789-213532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:06:27Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Aguero, Dan 2026-02-18T11:26:43Z 2025 On Complex Lie Algebroids with Constant Real Rank. Dan Aguero. SIGMA 21 (2025), 044, 25 pages 1815-0659 2020 Mathematics Subject Classification: 53D20 arXiv:2401.05274 https://nasplib.isofts.kiev.ua/handle/123456789/213532 https://doi.org/10.3842/SIGMA.2025.044 We associate a real distribution to any complex Lie algebroid that we call the distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles. The author acknowledges Henrique Bursztyn for proposing the topic and his many valuable observations. The author is also grateful to Hudson Lima, Roberto Rubio, and Pedro Frejlich for many fruitful and helpful conversations. We are also thankful to the anonymous referees for their valuable comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Complex Lie Algebroids with Constant Real Rank Article published earlier |
| spellingShingle | On Complex Lie Algebroids with Constant Real Rank Aguero, Dan |
| title | On Complex Lie Algebroids with Constant Real Rank |
| title_full | On Complex Lie Algebroids with Constant Real Rank |
| title_fullStr | On Complex Lie Algebroids with Constant Real Rank |
| title_full_unstemmed | On Complex Lie Algebroids with Constant Real Rank |
| title_short | On Complex Lie Algebroids with Constant Real Rank |
| title_sort | on complex lie algebroids with constant real rank |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213532 |
| work_keys_str_mv | AT aguerodan oncomplexliealgebroidswithconstantrealrank |