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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

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