Liviu Popescu
Abstract.In this paper we generalize the linear contravariant connection on Poisson manifolds to Lie algebroids and study its tensors of torsion and curvature. A Poisson connection which depends only on the Poisson bivector and structural functions of the Lie algebroid is given. The notions of complete and horizontal lifts are introduced and their compatibility conditions are pointed out.
M.S.C. 2000: 53D17, 17B66, 53C05.
Key words: Poisson manifolds, Lie algebroids, contravariant connection, complete and horizontal lifts.
1
Introduction
Poisson manifolds were introduced by A. Lichnerowicz in his famous paper [11] and their properties were later investigated by A. Weinstein [19]. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. The Lie algebroid [12] is a generalization of a Lie algebra and integrable distribution. In fact, a Lie algebroid is a vector bundle with a Lie bracket on his space of sections whose properties are very similar to those of a tangent bundle. We remark that the cotangent bundle of a Poisson manifold has a natural structure of a Lie algebroid [18]. In the last years the various aspects of these subjects have been studied in the different directions of research ([18], [15], [3], [4] [13], [10], [1], [9], [2]). In [16] the author has investigated the properties of connections on a Lie algebroid and together with D. Hrimiuc [8] studied the nonlinear connections of its dual. In [3] the linear contravariant Poisson connections on vector bundle are pointed out.
The purpose of this paper is to study some aspects of the geometry of the Lie alge-broids endowed with a Poisson structure, which generalize some results on Poisson manifolds to Lie algebroids. The paper is organized as follows. In the section 2 we recall the Cartan calculus and the Schouten-Nijenhuis bracket to the level of a Lie algebroids and introduce the Poisson structure on Lie algebroid. We investigate the properties of linear contravariant connection and its tensors of torsion and curvature. In the last part of this section we find a Poisson connection which depends only on the Poisson bivector and structural functions of Lie algebroid which generalize the
∗
Balkan Journal of Geometry and Its Applications, Vol.14, No.2, 2009, pp. 79-89. c
results of R. Fernandes from [3].
The section 3 deals with the prolongation of Lie algebroid [7] over the vector bundle projection. We study the properties of the complete lift of a Poisson bivector and introduce the notion of horizontal lift. The compatibility conditions of these bivec-tors are investigated. We remark that in the particular case of the Lie algebroid (E=T M, σ=Id) some results of G. Mitric and I. Vaisman [15] are obtained.
2
Lie algebroids
Let us consider a differentiable, n-dimensional manifold M and (T M, πM, M) its
tangent bundle. A Lie algebroid over the manifoldM is the triple (E,[·,·], σ) where
π:E →M is a vector bundle of rankm overM, whoseC∞(M)-module of sections Γ(E) is equipped with a Lie algebra structure [·,·] andσ:E→T M is a bundle map (calledthe anchor) which induces a Lie algebra homomorphism (also denotedσ) from Γ(E) toχ(M), satisfying the Leibnitz rule
[s1, f s2] =f[s1, s2] + (σ(s1)f)s2,
for everyf ∈C∞(M) ands1,s
2 ∈Γ(E). Therefore, we get
[σ(s1), σ(s2)] =σ[s1, s2], [s1,[s2, s3]] + [s2,[s3, s1]] + [s3,[s1, s2]] = 0.
Ifω ∈Vk(E∗) then theexterior derivativedEω∈Vk+1
(E∗) is given by the formula
dEω(s
1, ..., sk+1) = k+1
X
i=1
(−1)i+1σ(s
i)ω(s1, ..., ˆ
si, ..., sk+1) +
+ X
1≤i<j≤k+1
(−1)i+jω([s
i,sj], s1, ..., ˆ
si, ...,
ˆ
sj, ...sk+1).
wheresi ∈Γ(E),i = 1, k+ 1, and it results that (dE)2 = 0. Also, forξ ∈Γ(E) one
can define theLie derivativewith respect to ξby
Lξ =iξ◦dE+dE◦iξ,
whereiξ is the contraction with ξ.
If we take the local coordinates (xi) on an openU ⊂M, a local basis{s
α}of sections of
the bundleπ−1(U)→U generates local coordinates (xi, yα) onE. The local functions
σi α(x),L
γ
αβ(x) onM given by
σ(sα) =σiα
∂
∂xi, [sα, sβ] =L γ
αβsγ, i= 1, n, α, β, γ= 1, m,
are called thestructure functionsof the Lie algebroid and satisfy thestructure equa-tionson Lie algebroid
σjα
∂σi β
∂xj −σ j β
∂σi α
∂xj =σ i γL
γ αβ,
X
(α,β,γ)
Ã
σiα
∂Lδ βγ
∂xi +L δ αηL
η βγ
!
Locally, if f ∈ C∞(M) then dEf = ∂f
∂xiσiαsα, where {sα} is the dual basis of {sα}
and ifθ∈Γ(E∗), θ=θ
αsαthen
dEθ= (σαi
∂θβ
∂xi −
1 2θγL
γ αβ)s
α∧sβ,
Particularly, we get
dExi=σαisα, dEsα=−
1 2L
α
βγsβ∧sγ.
The Schouten-Nijenhuis bracket onE is given by [18]
[X1∧...∧Xp, Y1∧...∧Yq] = (−1)p+1 p
X
i=1
q
X
j=1
(−1)i+j[X
i, Yj]∧X1∧...∧
ˆ
Xi∧...∧Xp∧ ∧Y1∧...∧ ˆ
Yj∧...∧Yq
whereXi, Yj ∈Γ(E).
2.1
Poisson structures on Lie algebroids
Let us consider the bivector (i.e. contravariant, skew-symmetric, 2-section) Π ∈
Γ(∧2E) given by the expression
(2.1) Π = 1
2π
αβ(x)s α∧sβ.
Definition 2.1The bivector Π is a Poisson bivector onEif and only if the relation
[Π,Π] = 0,
is fulfilled.
Proposition 2.1Locally, the condition[Π,Π] = 0is expressed as
(2.2) X
(α,ε,δ) (παβσiβ
∂πεδ
∂xi +π
αβπγδLε βγ) = 0
If Π is a Poisson bivector then the pair (E,Π) is called aLie algebroid with Poisson structure. A corresponding Poisson bracket onM is given by
{f1, f2}= Π(dEf1, dEf2), f1, f2∈C∞(M).
We also have the bundle mapπ#:E∗→E defined by
π#ρ=iρΠ, ρ∈Γ(E∗).
Let us consider the bracket
whereLis Lie derivative andρ, θ∈Γ(E∗).With respect to this bracket and the usual
Lie bracket on vector fields, the mapσe:E∗→T M given by
e
σ=σ◦π#,
is a Lie algebra homomorphism
e
σ[ρ, θ]π = [σρ,e eσθ].
The bracket [., .]π satisfies also the Leibnitz rule
[ρ, f θ]π =f[ρ, θ]π+eσ(ρ)(f)θ,
and it results that (E∗,[., .]
π,eσ) is a Lie algebroid [13]. Next, we can define the
con-travariant exterior differentialdπ:Vk
(E∗)→Vk+1
(E∗) by
dπω(s
1, ..., sk+1) = k+1
X
i=1
(−1)i+1
e
σ(si)ω(s1, ..., ˆ
si, ..., sk+1) +
+ X
1≤i<j≤k+1
(−1)i+jω([si,sj]π, s1, ..., ˆ
si, ...,
ˆ
sj, ...sk+1).
Accordingly, we get the cohomology of Lie algebroidE∗ with the anchor
e
σ and the bracket [., .]π which generalize the Poisson cohomology of Lichnerowicz for Poisson
manifolds [11].
In the following we deal with the notion of contravariant connection on Lie algebroids, which generalize the similar notion on Poisson manifolds [18], [3].
Definition 2.2 If ρ, θ ∈ Γ(E∗) and Φ,Ψ ∈ Γ(E) then the linear contravariant
connection on a Lie algebroid is an application D : Γ(E∗)×Γ(E) → Γ(E) which
satisfies the relations: i) Dρ+θΦ =DρΦ +DθΦ,
ii) Dρ(Φ + Ψ) =DρΦ +DρΨ,
iii)Df ρΦ =f DρΦ,
iv)Dρ(fΦ) =f DρΦ +σe(ρ)(f)Φ, f ∈C∞(M).
Definition 2.3The torsion and curvature of the linear contravariant connection are given by
T(ρ, θ) =Dρθ−Dθρ−[ρ, θ]π,
R(ρ, θ)µ=DρDθµ−DθDρµ−D[ρ,θ]πθ,
whereρ, θ, µ∈Γ(E∗).
In the local coordinates we define the Christoffel symbols Γαβ
γ considering
Dsαsβ= Γαβγ sγ,
and under a change of coordinatesxi′ =xi′
(xi),i, i′ = 1, nonM, and yα′ =Aα′
αyα,
α, α′ = 1, monE,corresponding to a new basesα′ =Aα′
αsα, these symbols transform
(2.3) Γαγ′′β′ =Aα
Proposition 2.2 The local components of the torsion and curvature of the linear contravariant connection on a Lie algebroid have the expressions
Tαβ
The contravariant connection induces a contravariant derivativeDα : Γ(E)→Γ(E)
such that the following relations are fulfilled
Df1α1+f2α2=f1Dα1+f2Dα2, fi∈C∞(M), αi∈Γ(E∗),
Dρ(fΦ) =f DρΦ +eσ(ρ)(f)Φ, f ∈C∞(M), ρ, θ∈Γ(E∗).
LetT be a tensor of type (r, s) with the componentsTi1...ir
j1...js and θ=θαs
α a section
ofE∗. The local coordinates expression of the contravariant derivative is given by
DθT =θαTj1...ji1...isr/
and/denote thecontravariat derivative operator.
We recall that a tensor fieldT onE is calledparallelif and only ifDT = 0.
Definition 2.4 A contravariant connection D is called a Poisson connection if the Poisson bivector is parallel with respect toD.
Let us consider a contravariant connection D with the Cristoffel symbols Γαβ γ and
associated contravariant derivative. We obtain:
Proposition 2.3The contravariant connectionD with the coefficients given by
(2.4) Γαβγ = Γαβ
γ −
1 2πγεπ
αε/β,
is a Poisson connection.
Proof. Considering / the contravariant derivative operator with respect to the contravariant connectionD, we get
=παεσi ε
∂πβγ
∂xi +
µ
Γβα ε −
1 2πετπ
βτ/α
¶
πεγ+
µ
Γγα ε −
1 2πετπ
γτ/α
¶
πβε= 0,
which concludes the proof. ⊓⊔
Remark 2.1Considering the expression
(2.5) Γαβγ =σγi
∂παβ
∂xi ,
in (2.4) we obtain a Poisson connectionD with the coefficients
Γαβγ =σγi
∂παβ
∂xi −
1 2πγεπ
αε/β,
which depends only on the Poisson bivector and structural functions of the Lie alge-broid.
Proof. Under a change of coordinates, the structure functionsσi
α change by the
rule [17], [14]
σi′
α′Aα ′
α =
∂xi′
∂xiσ i α,
and, by direct computation, follows that the coefficients (2.5) satisfy the
transforma-tion law (2.3). ⊓⊔
Theorem 2.1If the following relation
X
(α,ε,δ)
παβπγδLεβγ= 0,
is true, then the connectionD with the coefficients
Γαβ γ =σγi
∂παβ
∂xi ,
is a Poisson connection on Lie algebroid.
Proof. Using relation (2.2) we obtain thatπβγ/α = 0 if and only if the required
relation is fulfilled. ⊓⊔
Proposition 2.4The set of Poisson connections on Lie algebroid is given by
Γαβγ = Γαβ
γ + ΩαεγνXενβ,
where
Ωαεγν =
1 2
¡
δναδγε−πγνπαε
¢
,
andD(Γαβ
γ )is a Poisson connection withXεδβ an arbitrary tensor.
πβγ/α=παεσi
3
The prolongation of Lie algebroid over the vector
bundle projection
new Lie algebroid over manifoldE. An element ofTEis said to be vertical if it is in the kernel of the projectionπ1. We will denote (VTE, π2|VTE, E) the vertical bundlea unique vector field sc on E, the complete lift of s satisfying the two following
conditions:
i)sc isπ-projectable onσ(s),
ii)sc(∧α) =Ld sα,
for allα∈Γ(E∗),whereα∧(u) =α(π(u))(u),u∈E (see [5] [6]).
Considering the prolongationTE ofE over the projection π, we may introduce the
vertical lift sv and the complete lift sc of a section s ∈ Γ(E) as the sections of
TE →Egiven by (see [14])
sv(u) = (0, sv(u)), sc(u) = (s(π(u)), sc(u)), u∈E.
Another canonical object onTEis theEuler sectionC, which is the section ofTE→
and (∂/∂xi, ∂/∂yα) is the local basis onT E.The structure functions ofTEare given
by the following formulas
σ1(Xα) =σiα
∂ ∂xi, σ
1(V
α) =
∂ ∂yα,
[Xα,Xβ] =LγαβXγ, [Xα,Vβ] = 0, [Vα,Vβ] = 0.
The vertical lift of a sectionρ=ραs
αand the corresponding vector field areρv =ραVα
andσ1(ρv) =ρα ∂
∂yα. The expression of the complete lift of a sectionρis
ρc=ραX α+ (
·
ρα−Lα
βγρβyγ)Vα,
and therefore
σ1(ρc) =ρασiα
∂ ∂xi + (σ
i γ
∂ρα
∂xi −L α βγρβ)yγ
∂ ∂yα.
In particular
sv
α=Vα, scα=Xα−LβαγyγVβ.
The coordinate expressions ofCandσ1(C) are
C=yαV
α, σ1(C) =yα
∂ ∂yα.
The local expression of the differential of a functionL onTE isdEL=σi
α∂x∂LiXα+
∂L
∂yαVα,where{Xα,Vα} denotes the corresponding dual basis of{Xα,Vα} and
there-fore, we havedExi=σi
αXα anddEyα=Vα. The differential of sections of (TE)∗ is
determined by
dEXα=−1
2L
α
βγXβ∧ Xγ, dEVα= 0.
A nonlinear connection N on TE [17] is an m−dimensional distribution (called
horizontal distribution) N : u ∈ E → HTuE ⊂ TE that is supplementary to
the vertical distribution. This means that we have the following decomposition
TuE = HTuE ⊕VTuE, for u ∈ E. A connection N on TE induces two
projec-tors h,v : TE → TE such that h(ρ) =ρh and v(ρ) =ρv for every ρ∈Γ(TE). We have
h = 1
2(id+N), v = 1
2(id−N). The sections
δα= (Xα)h=Xα−NαβVβ,
generate a basis of HTE, where Nβ
α are the coefficients of nonlinear connection.
The frame{δα,Vα} is a local basis of TE called adapted. The dual adapted basis is
{Xα, δVα}where δVα=Vα−Nα
βXβ. The Lie brackets of the adapted basis{δα,Vα}
are [16]
[δα, δβ] =Lγαβδγ+RγαβVγ, [δα,Vβ] =
∂Nγ α
∂yβ Vγ, [Vα,Vβ] = 0,
(3.1) Rγαβ=δβ(Nαγ)−δα(Nβγ) +LεαβNεγ.
The curvature of a connectionN onTE is given by Ω =−Nh where h is horizontal projector and Nh is the Nijenhuis tensor of h. In the local coordinates we have
Ω =−1
where Rγαβ are given by (3.1) and represent the local coordinate functions of the curvature tensor Ω in the frameV2TE∗⊗ TE induced by{X
α,Vα}.
3.1
Compatible Poisson structures
Let us consider the Poisson bivector on Lie algebroid given by relation (2.1). We ob-tain:
Proposition 3.1The complete lift ofΠon TE is given by
(3.2) Πc=παβXα∧ Vβ+
Proof. Using the properties of vertical and complete lifts we obtain
Πc= (1 Proposition 3.2The complete liftΠc is a Poisson bivector onTE.
Proof. Using the relations (3.2) and (2.2), by straightforward computation, we
obtain that [Πc,Πc] = 0, which ends the proof. ⊓⊔
Proposition 3.3 The Poisson structure Πc has the following property
Πc=−LCΠc,
which means that(TE,Πc) is a homogeneous Poisson manifold.
Proof. A direct computation in local coordinates yields
⊓ ⊔ Definition 3.1Let us consider a Poisson bivector on E given by (2.1) then the horizontal lift of Π toTE is the bivector defined by
ΠH= 1
2π
αβ(x)δ α∧δβ.
Proposition 3.4 The horizontal liftΠH is a Poisson bivector if and only ifΠ is
a Poisson bivector onE and the following relation
παβπγδRε βγ= 0,
is fulfilled.
Proof. The Poisson condition [Π,Π] = 0 leads to the relation (2.2) and equation [ΠH,ΠH] = 0 yields
X
(ε,δ,α)
µ
παβπγδLεβγ+παβσiβ
∂πεδ
∂xi
¶
δε∧δα∧δδ+παβπγδRεβγVε∧δα∧δγ= 0,
which ends the proof. ⊓⊔
We recall that two Poisson structures are called compatible if the bivectorsω1 and
ω2 satisfy the condition
[ω1, ω2] = 0.
By straightforward computation in local coordinates we get:
Proposition 3.5 The Poisson bivector ΠH is compatible with the complete lift
Πc if and only if the following relations hold
πrβπαs(∂Nrγ
∂ys −
∂Nγ s
∂yr )−π
rγπsαLβ sr= 0,
πrs
µ
δr(aαβ)−alα
∂Nβ r
∂yl +a lβ∂Nrα
∂yl −
−πθβRα
rθ+ (πεβLθεγ−πεθLβεγ)yγ
∂Nα r
∂yθ +
+σi r
∂πεβ
∂xi y γLα
εγ+πεβLαεγNrγ−πεβσir
∂Lα εγ
∂xi y γ
¶
= 0.
where we have denoted
aαβ=σiε
∂παβ
∂xi y ε+Nα
επεβ−Nεβπεα.
References
[1] M. Anastasiei,Geometry of Lagrangians and semisprays on Lie algebroids, Proc. Conf. Balkan Society of Geometers, Mangalia, 2005, BSGP, (2006), 10-17. [2] Carlo Cattani,Harmonic wavelet solution of Poisson’s problem, Balkan Journal
of Geometry and Its Applications, 13, 1 (2008), 27-37.
[3] R. L. Fernandes,Connections in Poisson geometry I: holonomy and invariants, J. Differential Geometry, 54 (2000), 303-365.
[4] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Advances in Mathematics, 170 (2002), 119-179.
[5] J. Grabowski, P. Urbanski, Tangent and cotangent lift and graded Lie algebra associated with Lie algebroids, Ann. Global Anal. Geom. 15, (1997), 447-486. [6] J. Grabowski, P. Urbanski,Lie algebroids and Poisson-Nijenhuis structures, Rep.
Math. Phys., 40 (1997), 195-208.
[7] P. J. Higgins, K. Mackenzie,Algebraic constructions in the category of Lie alge-broids, Journal of Algebra 129 (1990), 194-230.
[8] D. Hrimiuc, L. Popescu, Nonlinear connections on dual Lie algebroids, Balkan Journal of Geometry and Its Applications, 11, 1 (2006), 73-80.
[9] M. Ivan, Gh. Ivan, D. Opris,The Maxwell-Block equations on fractional Leibniz algebroids, Balkan Journal of Geometry and Its Applications, 13, 2 (2008), 50-58. [10] M. de Leon, J. C. Marrero, E. Martinez,Lagrangian submanifolds and dynamics
on Lie algebroidsJ. Phys. A: Math. Gen. 38 (2005), 241-308.
[11] A. Lichnerowicz, Les vari´et´es de Poisson et leurs alg`ebres de Lie associ´ees, J. Differential Geometry, 12 (1977), 253-300.
[12] K. Mackenzie,Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Cambridge, 124, 1987.
[13] K. Mackenzie, Ping Xu, Lie bialgebroids and Poisson groupoids, Duke Math. Journal, 73, 2 (1994), 415-452.
[14] E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67, (2001), 295-320.
[15] G. Mitric, I. Vaisman, Poisson structures on tangent bundles, Diff. Geom. and Appl., 18 (2003), 207-228.
[16] L. Popescu,Geometrical structures on Lie algebroids, Publ. Math. Debrecen 72, 1-2 (2008), 95-109.
[17] P. Popescu, On the geometry of relative tangent spaces, Rev. Roumain Math. Pures and Applications, 37, 8 (1992), 727-733.
[18] I. Vaisman,Lectures on the geometry of Poisson manifolds, Progress in Math., 118, Birkh¨auser, Berlin, 1994.
[19] A. Weinstein,The local structure of Poisson manifolds,J. Differential Geometry, 18 (1983), 523-557
Author’s address:
Liviu Popescu
University of Craiova,
Dept. of Applied Mathematics in Economy 13 A.I.Cuza st., 200585, Craiova, Romania
