doi: 10.3934/jgm.2021029
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Poisson double structures

1. 

Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil

2. 

Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, CEP 21941-909, Rio de Janeiro, Brazil

3. 

Universidade Federal Fluminense (UFF), Rua Professor Marcos Waldemar de Freitas Reis, s/n, Niterói, 24.210-201 RJ, Brazil

*Corresponding author: Henrique Bursztyn

To the memory of Kirill Mackenzie

Received  June 2021 Early access December 2021

We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.

Citation: Henrique Bursztyn, Alejandro Cabrera, Matias del Hoyo. Poisson double structures. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021029
References:
[1]

D. Álvarez, Leaves of stacky Lie algebroids, Comptes Rendus. Mathématique, 358 (2020), 217-226.  doi: 10.5802/crmath.37.

[2]

D. Álvarez, Poisson groupoids and moduli spaces of flat bundles over surfaces, arXiv: 2106.11078.

[3]

J. C. Baez and A. S. Crans, Higher-dimensional algebra. Ⅵ. Lie 2-algebras, Theory Appl. Categ., 12 (2004), 492-538. 

[4]

C. BaiY. Sheng and C. Zhu, Lie 2-bialgebras, Comm. Math. Phys., 320 (2013), 149-172.  doi: 10.1007/s00220-013-1712-3.

[5]

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.

[6]

R. Brown and C. B. Spencer, G-groupoids, crossed modules, and the classifying space of a topological group, Indagationes Mathematicae (Proceedings), 79 (1976), 296-302.  doi: 10.1016/1385-7258(76)90068-8.

[7]

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

[8]

A. CabreraI. Mǎrcuţ and M. A. Salazar, On local integration of Lie brackets, J. Reine Angew. Math. (Crelle's journal), 760 (2020), 267-293.  doi: 10.1515/crelle-2018-0011.

[9]

A. S. Cattaneo, On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds, Lett. Math. Phys., 67 (2004), 33-48.  doi: 10.1023/B:MATH.0000027690.76935.f3.

[10]

Z. ChenM. Stiénon and P. Xu, Poisson 2-groups, J. Differential Geom., 94 (2013), 209-240.  doi: 10.4310/jdg/1367438648.

[11]

Z. ChenM. Stiénon and P. Xu, Weak Lie 2-bialgebras, J. Geom. Phys., 68 (2013), 59-68.  doi: 10.1016/j.geomphys.2013.01.006.

[12]

N. Ciccoli, Quantization of co-isotropic subgroups, Lett. Math. Phys., 42 (1997), 123-138.  doi: 10.1023/A:1007352218739.

[13]

V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang - Baxter equations, Soviet Math. Dokl., 27 (1983), 68-71. 

[14]

T. DrummondM. Jotz Lean and C. Ortiz, VB-algebroid morphisms and representations up to homotopy, Diff. Geom. Appl., 40 (2015), 332-357.  doi: 10.1016/j.difgeo.2015.03.005.

[15]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.

[16]

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.

[17]

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.

[18]

A. Gracia-Saz and R. A. Mehta, $\mathcal{VB}$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.  doi: 10.4310/JSG.2017.v15.n3.a5.

[19]

M. V. Karasev, Analogues of objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv., 28 (1987), 497-527.  doi: 10.1070/IM1987v028n03ABEH000895.

[20]

Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Mathematicae, 41 (1995), 153-165.  doi: 10.1007/BF00996111.

[21]

J.-H. Lu and A. Weinstein, Groupoïdes symplectiques doubles des groupes de Lie Poisson, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 951-954. 

[22]

J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing actions and Bruhat decompositions, J. Diff. Geom., 31 (1990), 501-526.  doi: 10.4310/jdg/1214444324.

[23]

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.

[24]

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

[25]

K. C. H. Mackenzie, On symplectic double groupoids and the duality of Poisson groupoids, Internat. J. Math., 10 (1999), 435-456.  doi: 10.1142/S0129167X99000185.

[26]

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

[27]

K. C. H. Mackenzie, Notions of double for Lie algebroids, arXiv: math/0011212.

[28]

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

[29]

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.

[30]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.

[31]

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

[32]

R. A. Mehta, Q-algebroids and their cohomology, J. Symplectic Geom., 7 (2009), 263-293.  doi: 10.4310/JSG.2009.v7.n3.a1.

[33]

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.

[34]

I. Moerdijk and J. Mrcun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115.  doi: 10.1016/j.aim.2005.05.011.

[35]

T. Mokri, Matched pairs of Lie algebroids, Glasgow Math. J., 39 (1997), 167-181.  doi: 10.1017/S0017089500032055.

[36]

B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homology, Homotopy Appl., 9 (2007), 75-106.  doi: 10.4310/HHA.2007.v9.n1.a3.

[37]

L. Stefanini, On the integration of LA-groupoids and duality for Poisson groupoids, Travaux Mathématiques, 17 (2007), 65-85. 

[38]

L. Stefanini, On Morphic Actions and Integrability of LA-Groupoids, PhD. Thesis, Univ. Zurich, 2009. arXiv: 0902.2228.

[39]

A. Yu Vaintrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), 161-162.  doi: 10.1070/RM1997v052n02ABEH001802.

[40]

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

[41]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101-104.  doi: 10.1090/S0273-0979-1987-15473-5.

[42]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727.  doi: 10.2969/jmsj/04040705.

show all references

References:
[1]

D. Álvarez, Leaves of stacky Lie algebroids, Comptes Rendus. Mathématique, 358 (2020), 217-226.  doi: 10.5802/crmath.37.

[2]

D. Álvarez, Poisson groupoids and moduli spaces of flat bundles over surfaces, arXiv: 2106.11078.

[3]

J. C. Baez and A. S. Crans, Higher-dimensional algebra. Ⅵ. Lie 2-algebras, Theory Appl. Categ., 12 (2004), 492-538. 

[4]

C. BaiY. Sheng and C. Zhu, Lie 2-bialgebras, Comm. Math. Phys., 320 (2013), 149-172.  doi: 10.1007/s00220-013-1712-3.

[5]

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.

[6]

R. Brown and C. B. Spencer, G-groupoids, crossed modules, and the classifying space of a topological group, Indagationes Mathematicae (Proceedings), 79 (1976), 296-302.  doi: 10.1016/1385-7258(76)90068-8.

[7]

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

[8]

A. CabreraI. Mǎrcuţ and M. A. Salazar, On local integration of Lie brackets, J. Reine Angew. Math. (Crelle's journal), 760 (2020), 267-293.  doi: 10.1515/crelle-2018-0011.

[9]

A. S. Cattaneo, On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds, Lett. Math. Phys., 67 (2004), 33-48.  doi: 10.1023/B:MATH.0000027690.76935.f3.

[10]

Z. ChenM. Stiénon and P. Xu, Poisson 2-groups, J. Differential Geom., 94 (2013), 209-240.  doi: 10.4310/jdg/1367438648.

[11]

Z. ChenM. Stiénon and P. Xu, Weak Lie 2-bialgebras, J. Geom. Phys., 68 (2013), 59-68.  doi: 10.1016/j.geomphys.2013.01.006.

[12]

N. Ciccoli, Quantization of co-isotropic subgroups, Lett. Math. Phys., 42 (1997), 123-138.  doi: 10.1023/A:1007352218739.

[13]

V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang - Baxter equations, Soviet Math. Dokl., 27 (1983), 68-71. 

[14]

T. DrummondM. Jotz Lean and C. Ortiz, VB-algebroid morphisms and representations up to homotopy, Diff. Geom. Appl., 40 (2015), 332-357.  doi: 10.1016/j.difgeo.2015.03.005.

[15]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.

[16]

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.

[17]

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.

[18]

A. Gracia-Saz and R. A. Mehta, $\mathcal{VB}$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.  doi: 10.4310/JSG.2017.v15.n3.a5.

[19]

M. V. Karasev, Analogues of objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv., 28 (1987), 497-527.  doi: 10.1070/IM1987v028n03ABEH000895.

[20]

Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Mathematicae, 41 (1995), 153-165.  doi: 10.1007/BF00996111.

[21]

J.-H. Lu and A. Weinstein, Groupoïdes symplectiques doubles des groupes de Lie Poisson, C. R. Acad. Sci. Paris Sér. I Math., 309 (1989), 951-954. 

[22]

J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing actions and Bruhat decompositions, J. Diff. Geom., 31 (1990), 501-526.  doi: 10.4310/jdg/1214444324.

[23]

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.

[24]

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

[25]

K. C. H. Mackenzie, On symplectic double groupoids and the duality of Poisson groupoids, Internat. J. Math., 10 (1999), 435-456.  doi: 10.1142/S0129167X99000185.

[26]

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

[27]

K. C. H. Mackenzie, Notions of double for Lie algebroids, arXiv: math/0011212.

[28]

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

[29]

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.

[30]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.

[31]

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

[32]

R. A. Mehta, Q-algebroids and their cohomology, J. Symplectic Geom., 7 (2009), 263-293.  doi: 10.4310/JSG.2009.v7.n3.a1.

[33]

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.

[34]

I. Moerdijk and J. Mrcun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115.  doi: 10.1016/j.aim.2005.05.011.

[35]

T. Mokri, Matched pairs of Lie algebroids, Glasgow Math. J., 39 (1997), 167-181.  doi: 10.1017/S0017089500032055.

[36]

B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homology, Homotopy Appl., 9 (2007), 75-106.  doi: 10.4310/HHA.2007.v9.n1.a3.

[37]

L. Stefanini, On the integration of LA-groupoids and duality for Poisson groupoids, Travaux Mathématiques, 17 (2007), 65-85. 

[38]

L. Stefanini, On Morphic Actions and Integrability of LA-Groupoids, PhD. Thesis, Univ. Zurich, 2009. arXiv: 0902.2228.

[39]

A. Yu Vaintrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), 161-162.  doi: 10.1070/RM1997v052n02ABEH001802.

[40]

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

[41]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101-104.  doi: 10.1090/S0273-0979-1987-15473-5.

[42]

A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40 (1988), 705-727.  doi: 10.2969/jmsj/04040705.

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