doi: 10.3934/jgm.2021023
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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 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, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

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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