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On local integration of Lie brackets

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Veröffentlicht/Copyright: 9. Mai 2018
Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2020 Heft 760

Abstract

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

Funding statement: Ioan Mărcuţ was supported by the NWO Veni grant 613.009.031 and the NSF grant DMS 14-05671. María Amelia Salazar was a Post-Doctorate at IMPA during part of this project, funded by CAPES-Brazil. Alejandro Cabrera would like to thank CNPq and FAPERJ for financial support.

A Differentiating integrated cochains

In this appendix, we consider the differentiation of the local Lie groupoid cochains obtained by integration through the map Ψ defined in Section 2.5. The main result is Lemma A.1 which is used in the proof of item (ii) of Proposition 2.10.

Following the notation of Section 2.5, let α∈Γ⁢(∧pA*) be an algebroid p-cochain and let Ψ⁢(α):G(p)⇢ℝ be the corresponding local groupoid cochain. In order to apply the van Est map, we consider sections a1,…,ak∈Γ⁢(A) and compute the iterated differentiation Dap⁢⋯⁢Da1⁢Ψ⁢(α). We do this inductively on the number p of derivatives.

Let us fix composable arrows g2,…,gp∈G(p-1) in a sufficiently small neighborhood of M, and consider h1⁢(ϵ)=ϵ⁢a1⁢(τ⁢(g2)) so that

Da1⁢Ψ⁢(α)⁢(g2,…,gp)
  =dd⁢ϵ|ϵ=0⁢Ψ⁢(α)⁢(h1⁢(ϵ),g2,…,gp)
  =dd⁢ϵ|ϵ=0⁢∫Ip(θ*⁢α)⁢(∂1⁡γh1⁢(ϵ),g2,…,gp,…,∂p⁡γh1⁢(ϵ),g2,…,gp)⁢𝑑t1⁢…⁢𝑑tp.

By using the relations

(A.1)γϵ⁢g1,g2,…,gp⁢(t1,…,tp)=γg1,…,gp⁢(ϵ⁢t1,t2,…,tp),
γτ⁢(g2),g2,…,gp⁢(t1,…,tp)=γg2,…,gp⁢(t2,…,tp),

where τ⁢(g2)=σ⁢(g1)=0⋅g1, it follows that, for a∈Γ⁢(A) and small (g2,…,gn)∈G(n),

(A.2)∂1⁡γϵ⁢a⁢(τ⁢g2),g2,…,gn⁢(t1,…,tn)=ϵ⋅∂1⁡γa⁢(τ⁢(g2)),g2,…,gn⁢(ϵ⁢t1,t2,…,tn),
∂j⁡γτ⁢(g2),g2,…,gn⁢(t1,t2,…,tn)=∂j-1⁡γg2,…,gn⁢(t2,…,tn),j>1.

Applying (A.2), we can compute dd⁢ϵ|ϵ=0 (equivalently, apply the substitution u=ϵ⁢t1 as in Example 2.9) yielding

Da1Ψ(α)(g2,…,gp)=∫Ip-1(θ*α)(∂1γa1⁢(τ⁢(g2)),g2,…,gp(0,t2,…,tp),
∂1γg2,…,gp(t2,…,tp),…,∂p-1γg2,…,gp(t2,…,tp))dt2…dtp,

where ∫I𝑑t1=1 has been factored out. Above, the first factor inside α is of a different nature from the rest: it is the only one depending on a1 while the others only depend on the fixed string g2,…,gp. We introduce the following general notation for such terms: given a∈Γ⁢(A), small (k1,…,kn)∈G(n) and any (s1,…,sn)∈In,

(A.3)vk1,…,kna⁢(s1,…,sn):=∂1⁡γa⁢(τ⁢(k1)),k1,…,kn⁢(0,s1,…,sn);

thus, we have

vk1,…,kna⁢(s1,…,sn)∈Tσ⁢Gγk1,…,kn⁢(s1,…,sn).

Before stating the general inductive result, let us compute one more derivative fixing elements g3,…,gp∈G(p-2) and letting h2⁢(ϵ)=ϵ⁢a2⁢(τ⁢(g3)),

Da2⁢Da1⁢Ψ⁢(α)⁢(g3,…,gp)=dd⁢ϵ|ϵ=0⁢Da1⁢Ψ⁢(α)⁢(h2⁢(ϵ),g3,…,gp)
=dd⁢ϵ|ϵ=0∫Ip-1(θ*α)(vh2⁢(ϵ),g3,…,gpa1,∂1γh2⁢(ϵ),g3,…,gp,…,
∂p-1γh2⁢(ϵ),g3,…,gp)(t1,…,tp-1)dt1…dtp-1,

where we have relabeled the integration variables. Using (A.2) we compute dd⁢ϵ|ϵ=0, yielding the integral over d⁢t1⁢…⁢d⁢tp-1 of

(θ*α)(vh2⁢(0),g3,…,gpa1(t1,…,tp-1),vg3,…,gpa2(t2,…,tp-1),
∂1γg3,…,gp(t2,…,tp-1),…,∂p-2γg3,…,gp(t2,…,tp-1)).

The key step for recognizing an inductive structure in our computation is to note that (recall that M is identified with the unit section)

(A.4)γg1,τ⁢(g3),g3,…,gn⁢(t1,t2,…,tn)=γg1,g3,…,gn⁢(t1⋅t2,t3,…,tn)

for any small composable string (g1,g3,g4,…,gn)∈G(n-1) and any (t1,t2,…,tn)∈In. Using (A.4) and the notation (A.3), we get

(A.5)vτ⁢(g3),g3,…,gna⁢(t1,…,tn-1)=t1⋅vg3,…,gna⁢(t2,…,tn-1)

for any a∈Γ⁢(A), small (g3,…,gn)∈G(n-2) and (t1,…,tn-1)∈In-1. Thus, since

h2⁢(0)=τ⁢(g3),

we have

Da2Da1Ψ(α)(g3,…,gp)=∫Ip-1t1⋅(θ*α)(vg3,…,gpa1,vg3,…,gpa2,
∂1γg3,…,gp,…,∂p-2γg3,…,gp)(t2,…,tp-1)dt1…dtp-1

from which ∫It1⁢𝑑t1=12 factors out yielding

Da2⁢Da1⁢Ψ⁢(α)⁢(g3,…,gp)
  =12⁢∫Ip-2(θ*⁢α)⁢(vg3,…,gpa1,vg3,…,gpa2,∂1⁡γg3,…,gp,…,∂p-2⁡γg3,…,gp).

Continuing by induction, we obtain the following:

Lemma A.1.

For a1,…,ak∈Γ⁢(A) and (gk+1,…,gp)∈G(p-k) small enough, we have

(A.6)Dak⁢⋯⁢Da1⁢Ψ⁢(α)⁢(gk+1,…,gp)
 =1k!∫Ip-k(θ*α)(vgk+1,…,gpa1,…,vgk+1,…,gpak,
∂1γgk+1,…,gp,…,∂p-kγgk+1,…,gp),

where vgk+1,…,gpa:Ip-k→Tσ⁢G is defined in (A.3).

Proof.

We will consider an induction over k≥1 recalling that the case k=1 was already worked out above. Assume now (A.6) and compute

Dak+1⁢⋯⁢Da1⁢Ψ⁢(α)⁢(gk+2,…,gp)
=dd⁢ϵ|ϵ=0⁢Dak⁢⋯⁢Da1⁢Ψ⁢(α)⁢(ϵ⁢ak+1⁢(τ⁢(gk+2)),gk+2,…,gp).

To use (A.6) on the right-hand side above, let us introduce the following notation:

wgk+1,…,gpa1,…,ak:=vgk+1,…,gpa1∧…∧vgk+1,…,gpak

and b:=ak+1⁢(τ⁢(gk+2)), so that

(A.7)k!⁢Dak+1⁢⋯⁢Da1⁢Ψ⁢(α)⁢(gk+2,…,gp)
 =dd⁢ϵ|ϵ=0⁢∫Ip-k(θ*⁢α)⁢(wϵ⁢b,gk+2,…,gpa1,…,ak∧d⁢γϵ⁢b,gk+2,…,gp).

By using (A.1), it follows that

d⁢γϵ⁢b,gk+2,…,gp⁢(t1,…,tp-k)=ϵ⋅d⁢γb,gk+2,…,gp⁢(ϵ⁢t1,…,tp-k),

and that

γb,gk+2,…,gp⁢(0,t2,…,tp-k)=γgk+2,…,gp⁢(t2,…,tp-k).

Using these to compute (A.7) gives

k!⁢Dak+1⁢⋯⁢Da1⁢Ψ⁢(α)⁢(gk+2,…,gp)
=∫Ip-k(θ*⁢α)⁢(wτ⁢(gk+2),gk+2,…,gpa1,…,ak⁢(t1,…,tp-k)∧d⁢γgk+2,…,gp⁢(t2,…,tp-k))⁢𝑑t1⁢…⁢𝑑tp-k.

Applying (A.5), we get

wτ⁢(gk+2),gk+2,…,gpa1,…,ak⁢(t1,…,tp-k)=t1k⋅wgk+2,…,gpa1,…,ak⁢(t2,t3,…,tp-k).

The lemma thus follows by noticing that ∫01t1k⁢𝑑t1=1k+1 factors out. ∎

Acknowledgements

The authors would like to thank Marius Crainic, Pedro Frejlich, Rui Loja Fernandes, Marco Gualtieri, Eckhard Meinrenken and Daniele Sepe for useful discussions, and the anonymous referee for their careful reading and suggestions.

References

[1] R. Almeida and P. Molino, Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 1, 13–15. Suche in Google Scholar

[2] A. Cabrera, I. Mărcuţ and M. A. Salazar, Explicit formulas for multiplicative objects near the identities, in preparation. Suche in Google Scholar

[3] A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A 87, Université Claude-Bernard, Lyon (1987), 1–62. Suche in Google Scholar

[4] M.  Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681–721. 10.1007/s00014-001-0766-9Suche in Google Scholar

[5] M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. 10.4007/annals.2003.157.575Suche in Google Scholar

[6] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer, Berlin 2000. 10.1007/978-3-642-56936-4Suche in Google Scholar

[7] R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), no. 1, 119–179. 10.1006/aima.2001.2070Suche in Google Scholar

[8] R. L. Fernandes and I. Struchiner, The classifying Lie algebroid of a geometric structure I: Classes of coframes, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2419–2462. 10.1090/S0002-9947-2014-05973-4Suche in Google Scholar

[9] P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), no. 1, 194–230. 10.1016/0021-8693(90)90246-KSuche in Google Scholar

[10] D. Li-Bland and E. Meinrenken, On the van Est homomorphism for Lie groupoids, Enseign. Math. 61 (2015), no. 1–2, 93–137. 10.4171/LEM/61-1/2-5Suche in Google Scholar

[11] K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Math. Soc. Lecture Note Ser. 213, Cambridge University Press, Cambridge 2005. 10.1017/CBO9781107325883Suche in Google Scholar

[12] J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A907–A910. Suche in Google Scholar

[13] J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B 264 (1967), A245–A248. Suche in Google Scholar

[14] J. Pradines, Troisième théorème de Lie les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A21–A23. Suche in Google Scholar

[15] W. T. van Est, Rapport sur les S-atlas, Astérisque (1984), no. 116, 235–292. Suche in Google Scholar

[16] A. Weinstein and P. Xu, Extensions of symplectic groupoids and quantization, J. reine angew. Math. 417 (1991), 159–189. 10.1515/crll.1991.417.159Suche in Google Scholar

[17] O. Yudilevich, The role of the Jacobi identity in solving the Maurer–Cartan structure equation, Pacific J. Math. 282 (2016), no. 2, 487–510. 10.2140/pjm.2016.282.487Suche in Google Scholar

Received: 2017-04-06
Revised: 2018-04-05
Published Online: 2018-05-09
Published in Print: 2020-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  2. Probabilistic Schubert calculus
  3. On cohomological Hall algebras of quivers: Generators
  4. Intersections of two Grassmannians in ℙ9
  5. Erratum to Tropical mirror symmetry for elliptic curves (J. reine angew. Math. 732 (2017), 211–246)
  6. Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)
  7. Maximal time existence of unnormalized conical Kähler–Ricci flow
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  9. Undistorted purely pseudo-Anosov groups
  10. Dendroidal spaces, Γ-spaces and the special Barratt--Priddy--Quillen theorem
  11. On local integration of Lie brackets
https://doi.org/10.1515/crelle-2018-0011

Artikel in diesem Heft

  1. Frontmatter
  2. Probabilistic Schubert calculus
  3. On cohomological Hall algebras of quivers: Generators
  4. Intersections of two Grassmannians in ℙ9
  5. Erratum to Tropical mirror symmetry for elliptic curves (J. reine angew. Math. 732 (2017), 211–246)
  6. Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)
  7. Maximal time existence of unnormalized conical Kähler–Ricci flow
  8. A remark on a converse theorem of Cogdell and Piatetski-Shapiro
  9. Undistorted purely pseudo-Anosov groups
  10. Dendroidal spaces, Γ-spaces and the special Barratt--Priddy--Quillen theorem
  11. On local integration of Lie brackets
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