September  2021, 13(3): 403-457. doi: 10.3934/jgm.2021023

Transitive double Lie algebroids via core diagrams

1. 

Institute for Mathematics, Julius-Maximilians-Universität Würzburg, Germany

2. 

In memoriam, School of Mathematics and Statistics, The University of Sheffield, United Kingdom

*Corresponding author: Madeleine Jotz Lean

Received  March 2021 Revised  July 2021 Published  September 2021 Early access  August 2021

Fund Project: This research was a joint project with the sadly deceased second author. This paper is dedicated to his memory

The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.

The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.

Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.

Citation: Madeleine Jotz Lean, Kirill C. H. Mackenzie. Transitive double Lie algebroids via core diagrams. Journal of Geometric Mechanics, 2021, 13 (3) : 403-457. doi: 10.3934/jgm.2021023
References:
[1]

C. A. Abad and M. Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., 663 (2012), 91-126.  doi: 10.1515/CRELLE.2011.095.

[2]

I. Androulidakis, Crossed modules and the integrability of Lie brackets, preprint, arXiv: math/0501103.

[3]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[4]

H. BursztynA. Cabrera and M. del Hoyo, Vector bundles over Lie groupoids and algebroids, Adv. Math., 290 (2016), 163-207.  doi: 10.1016/j.aim.2015.11.044.

[5]

A. CabreraO. Brahic and C. Ortiz, Obstructions to the integrability of ${\mathcal {V B}}$-algebroids, J. Symplectic Geom., 16 (2018), 439-483.  doi: 10.4310/JSG.2018.v16.n2.a3.

[6]

T. DrummondM. Jotz Lean and C. Ortiz, ${\mathcal {V B}}$-algebroid morphisms and representations up to homotopy, Differential Geom. Appl., 40 (2015), 332-357.  doi: 10.1016/j.difgeo.2015.03.005.

[7]

A. Gracia-SazM. Jotz LeanK. C. H. Mackenzie and R. A. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

[8]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.  doi: 10.1016/j.aim.2009.09.010.

[9] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. 
[10]

E. Hawkins, A groupoid approach to quantization, J. Symplectic Geom., 6 (2008), 61-125.  doi: 10.4310/JSG.2008.v6.n1.a4.

[11]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665-699. 

[12]

P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230.  doi: 10.1016/0021-8693(90)90246-K.

[13]

M. Jotz Lean and C. Ortiz, Foliated groupoids and infinitesimal ideal systems, Indag. Math. (N.S.), 25 (2014), 1019-1053.  doi: 10.1016/j.indag.2014.07.009.

[14]

H.-Y. LiaoM. Stiénon and P. Xu, Formality and Kontsevich-Duflo type theorems for Lie pairs, Adv. Math., 352 (2019), 406-482.  doi: 10.1016/j.aim.2019.04.047.

[15]

S. Mac Lane, Categories for the Working Mathematician, $2^nd$ edition, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.

[16]

K. C. H. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, arXiv: math/9808081.

[17]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. I, Adv. Math., 94 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.

[18]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. II, Adv. Math., 154 (2000), 46-75.  doi: 10.1006/aima.1999.1892.

[19]

K. C. H. Mackenzie, Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids, J. Reine Angew. Math., 658 (2011), 193-245.  doi: 10.1515/CRELLE.2011.092.

[20] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[21]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467.  doi: 10.1016/S0040-9383(98)00069-X.

[22]

E. Meinrenken and J. Pike, The Weil algebra of a double Lie algebroid, Int. Math. Res. Not. IMRN, (2021), 8550–8622. doi: 10.1093/imrn/rnz361.

[23]

J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets Non Holonomes, Esquisses Mathématiques [Mathematical Sketches], 29, Université d'Amiens U.E.R. de Mathématiques, Amiens, 1977.

[24]

D. Quillen, Superconnections and the Chern character, Topology, 24 (1985), 89-95.  doi: 10.1016/0040-9383(85)90047-3.

[25]

L. Stefanini, On morphic actions and integrability of LA-groupoids, preprint, arXiv: 0902.2228.

[26]

M. Stiénon, L. Vitagliano and P. Xu, ${A}_\infty$-algebras from Lie pairs, work in progress.

[27]

T. T. Voronov, $Q$-manifolds and Mackenzie theory, Comm. Math. Phys., 315 (2012), 279-310.  doi: 10.1007/s00220-012-1568-y.

show all references

References:
[1]

C. A. Abad and M. Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., 663 (2012), 91-126.  doi: 10.1515/CRELLE.2011.095.

[2]

I. Androulidakis, Crossed modules and the integrability of Lie brackets, preprint, arXiv: math/0501103.

[3]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[4]

H. BursztynA. Cabrera and M. del Hoyo, Vector bundles over Lie groupoids and algebroids, Adv. Math., 290 (2016), 163-207.  doi: 10.1016/j.aim.2015.11.044.

[5]

A. CabreraO. Brahic and C. Ortiz, Obstructions to the integrability of ${\mathcal {V B}}$-algebroids, J. Symplectic Geom., 16 (2018), 439-483.  doi: 10.4310/JSG.2018.v16.n2.a3.

[6]

T. DrummondM. Jotz Lean and C. Ortiz, ${\mathcal {V B}}$-algebroid morphisms and representations up to homotopy, Differential Geom. Appl., 40 (2015), 332-357.  doi: 10.1016/j.difgeo.2015.03.005.

[7]

A. Gracia-SazM. Jotz LeanK. C. H. Mackenzie and R. A. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

[8]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.  doi: 10.1016/j.aim.2009.09.010.

[9] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. 
[10]

E. Hawkins, A groupoid approach to quantization, J. Symplectic Geom., 6 (2008), 61-125.  doi: 10.4310/JSG.2008.v6.n1.a4.

[11]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665-699. 

[12]

P. J. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra, 129 (1990), 194-230.  doi: 10.1016/0021-8693(90)90246-K.

[13]

M. Jotz Lean and C. Ortiz, Foliated groupoids and infinitesimal ideal systems, Indag. Math. (N.S.), 25 (2014), 1019-1053.  doi: 10.1016/j.indag.2014.07.009.

[14]

H.-Y. LiaoM. Stiénon and P. Xu, Formality and Kontsevich-Duflo type theorems for Lie pairs, Adv. Math., 352 (2019), 406-482.  doi: 10.1016/j.aim.2019.04.047.

[15]

S. Mac Lane, Categories for the Working Mathematician, $2^nd$ edition, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.

[16]

K. C. H. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, arXiv: math/9808081.

[17]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. I, Adv. Math., 94 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.

[18]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. II, Adv. Math., 154 (2000), 46-75.  doi: 10.1006/aima.1999.1892.

[19]

K. C. H. Mackenzie, Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids, J. Reine Angew. Math., 658 (2011), 193-245.  doi: 10.1515/CRELLE.2011.092.

[20] K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[21]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467.  doi: 10.1016/S0040-9383(98)00069-X.

[22]

E. Meinrenken and J. Pike, The Weil algebra of a double Lie algebroid, Int. Math. Res. Not. IMRN, (2021), 8550–8622. doi: 10.1093/imrn/rnz361.

[23]

J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets Non Holonomes, Esquisses Mathématiques [Mathematical Sketches], 29, Université d'Amiens U.E.R. de Mathématiques, Amiens, 1977.

[24]

D. Quillen, Superconnections and the Chern character, Topology, 24 (1985), 89-95.  doi: 10.1016/0040-9383(85)90047-3.

[25]

L. Stefanini, On morphic actions and integrability of LA-groupoids, preprint, arXiv: 0902.2228.

[26]

M. Stiénon, L. Vitagliano and P. Xu, ${A}_\infty$-algebras from Lie pairs, work in progress.

[27]

T. T. Voronov, $Q$-manifolds and Mackenzie theory, Comm. Math. Phys., 315 (2012), 279-310.  doi: 10.1007/s00220-012-1568-y.

Figure 1.  The core diagram of a double Lie algebroid
Figure 2.  Core diagram of Lie groupoids
Figure 3.  The transitive core diagram of a transitive double Lie algebroid
Figure 4.  A morphism of core diagrams
Figure 6.  The core diagram of align="right"
Figure 7.  Square of Lie algebroid morphisms
Figure 8.  Setting of Section 5, the transitive core diagram $ \mathcal C $
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