September  2021, 13(3): ⅰ-ⅸ. doi: 10.3934/jgm.2021025

Preface to special issue in memory of Kirill C. H. Mackenzie: Part I

1. 

Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis Athens, 157-84, Greece

2. 

Instituto Nacional de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil

3. 

Department of Mathematics, Statistics and Operational Research, Section of Mathematics, Faculty of Sciences, C/ Astrofísico Fco Sanchez, 38071 La Laguna, Tenerife Canary Islands, Spain

4. 

Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

Published  September 2021 Early access  September 2021

Citation: Iakovos Androulidakis, Henrique Bursztyn, Juan-Carlos Marrero, Alan Weinstein. Preface to special issue in memory of Kirill C. H. Mackenzie: Part I. Journal of Geometric Mechanics, 2021, 13 (3) : ⅰ-ⅸ. doi: 10.3934/jgm.2021025
References:
[1]

R. Almeida and P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 13-15. 

[2]

H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), 360-366.  doi: 10.1007/BF01209171.

[3]

P. Bressler, The first Pontryagin class, Compos. Math., 143 (2007), 1127-1163.  doi: 10.1112/S0010437X07002710.

[4]

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.

[5]

H. Bursztyn and T. Drummond, Lie theory of multiplicative tensors, Math. Ann., 375 (2019), 1489-1554.  doi: 10.1007/s00208-019-01881-w.

[6]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, In Quantization of Singular Symplectic Quotients, Progr. Math., Birkhäuser, Basel, 198 (2001), 61–93.

[7]

A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.

[8]

J. CortésM. de LeónJ. C. MarreroD. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558.  doi: 10.1142/S0219887806001211.

[9]

A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, In Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, Univ. Claude-Bernard, Lyon, 87 (1987), 1–62.

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.

[11]

M. de León, M. Epstein and V. Jimenez, Material Geometry: Groupoids in Continuum Mechanics, World Scientific, Signapore, 2021. doi: 10.1142/12168.

[12]

C. Debord and G. Skandalis, Blowup constructions for lie groupoids and a boutet de monvel type calculus, 2017.

[13]

V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR, 268 (1983), 285-287. 

[14]

C. Ehresmann, Catégories topologiques et catégories différentiables, In Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, (1959), 137–150.

[15]

B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom., 40 (1994), 213-238.  doi: 10.4310/jdg/1214455536.

[16]

M. K. Flari and K. C. H. Mackenzie, Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.  doi: 10.1007/s11005-018-1103-y.

[17]

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.

[18]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.

[19]

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.

[20]

M. Gualtieri, Generalized complex geometry, Ann. of Math., 174 (2011), 75-123.  doi: 10.4007/annals.2011.174.1.3.

[21]

A. Haefliger, Groupoïdes d'holonomie et classifiants, Transversal Structure of Foliations (Toulouse, 1982), 116 (1984), 70-97. 

[22]

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

[23]

P. J. Higgins and K. C. H. Mackenzie, Fibrations and quotients of differentiable groupoids, J. London Math. Soc., 42 (1990), 101-110.  doi: 10.1112/jlms/s2-42.1.101.

[24]

P. J. Higgins and K. C. H. Mackenzie, Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures, Math. Proc. Cambridge Philos. Soc., 114 (1993), 471-488.  doi: 10.1017/S0305004100071760.

[25]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157-216.  doi: 10.1023/B:MATH.0000027508.00421.bf.

[26]

Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids, In Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 213–233.

[27]

Z.-J. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  doi: 10.4310/jdg/1214459842.

[28]

K. C. H. Mackenzie, Rigid cohomology of topological groupoids, J. Austral. Math. Soc. Ser. A, 26 (1978), 277-301.  doi: 10.1017/S1446788700011794.

[29]

K. C. H. Mackenzie, Cohomology of Locally Trivial Groupoids and Lie Algebroids, PhD thesis, Monash University, 1979. doi: 10.1017/S0004972700011187.

[30]

K. C. H. Mackenzie, Infinitesimal theory of principal bundles, 50th ANZAAS Conference, Adelaide.

[31]

K. C. H. Mackenzie, Integrability obstructions for extensions of Lie algebroids, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), 29-52. 

[32] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, vol. 124 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.
[33]

K. C. H. Mackenzie, A note on Lie algebroids which arise from groupoid actions, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), 283-302. 

[34]

K. C. H. Mackenzie, Infinitesimal characterization of homogeneous bundles, Proc. Amer. Math. Soc., 103 (1988), 1271-1277.  doi: 10.1090/S0002-9939-1988-0955021-1.

[35]

K. C. H. Mackenzie, On extensions of principal bundles, Ann. Global Anal. Geom., 6 (1988), 141-163.  doi: 10.1007/BF00133036.

[36]

K. C. H. Mackenzie, Classification of principal bundles and Lie groupoids with prescribed gauge group bundle, J. Pure Appl. Algebra, 58 (1989), 181-208.  doi: 10.1016/0022-4049(89)90157-6.

[37]

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.

[38]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.

[39]

K. C. H. Mackenzie, Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids, Electron. Res. Announc. Amer. Math. Soc., 4 (1998), 74-87.  doi: 10.1090/S1079-6762-98-00050-X.

[40]

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.

[41]

K. C. H. Mackenzie, Affinoid structures and connections, In Poisson Geometry (Warsaw, 1998), Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 51 (2000), 175–186.

[42]

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

[43]

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

[44]

K. C. H. Mackenzie, A unified approach to Poisson reduction, Lett. Math. Phys., 53 (2000), 215-232.  doi: 10.1023/A:1011055510672.

[45]

K. C. H. Mackenzie, On certain canonical diffeomorphisms in symplectic and Poisson geometry, In Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 187–198. doi: 10.1090/conm/315/05480.

[46]

K. C. H. Mackenzie, Duality and triple structures, In The Breadth of Symplectic and Poisson Geometry, Progr. Math., Birkhäuser Boston, Boston, MA, 232 (2005), 455–481. doi: 10.1007/0-8176-4419-9_15.

[47]

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.

[48]

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.

[49]

K. C. H. Mackenzie, Proving the Jacobi identity the hard way, In Geometric Methods in Physics, Trends Math., Birkhäuser/Springer, Basel, (2013), 357–366.

[50]

K. C. H. Mackenzie, From symplectic groupoids to double structures, In Geometric, Algebraic and Topological Methods for Quantum Field Theory, World Sci. Publ., Hackensack, NJ, (2017), 192–219.

[51]

K. C. H. Mackenzie and T. Mokri, Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids, Geom. Dedicata, 77 (1999), 317-330.  doi: 10.1023/A:1005180403695.

[52]

K. C. H. Mackenzie, A. Odzijewicz and A. Sliżewska, Poisson geometry related to Atiyah sequences, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), 29pp. doi: 10.3842/SIGMA.2018.005.

[53]

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.

[54]

K. C. H. Mackenzie and P. Xu, Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford Ser., 49 (1998), 59-85.  doi: 10.1093/qmathj/49.1.59.

[55]

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

[56]

W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/68), 1-32.  doi: 10.1007/BF00276433.

[57]

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.

[58]

M. Stienon and P. Xu, Atiyah classes and kontsevich-duflo type theorem for dg manifolds, Banach Center Publications, 123.

[59]

S. Vassout, Raconte-moi... un groupoïde de Lie, Gazette des Mathématiciens, 147 (2016), 58-61. 

[60]

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

[61]

A. Weinstein, Lagrangian mechanics and groupoids, In Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 7 (1996), 207–231.

show all references

References:
[1]

R. Almeida and P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 13-15. 

[2]

H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), 360-366.  doi: 10.1007/BF01209171.

[3]

P. Bressler, The first Pontryagin class, Compos. Math., 143 (2007), 1127-1163.  doi: 10.1112/S0010437X07002710.

[4]

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.

[5]

H. Bursztyn and T. Drummond, Lie theory of multiplicative tensors, Math. Ann., 375 (2019), 1489-1554.  doi: 10.1007/s00208-019-01881-w.

[6]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, In Quantization of Singular Symplectic Quotients, Progr. Math., Birkhäuser, Basel, 198 (2001), 61–93.

[7]

A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.

[8]

J. CortésM. de LeónJ. C. MarreroD. Martín de Diego and E. Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 509-558.  doi: 10.1142/S0219887806001211.

[9]

A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, In Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, Univ. Claude-Bernard, Lyon, 87 (1987), 1–62.

[10]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.

[11]

M. de León, M. Epstein and V. Jimenez, Material Geometry: Groupoids in Continuum Mechanics, World Scientific, Signapore, 2021. doi: 10.1142/12168.

[12]

C. Debord and G. Skandalis, Blowup constructions for lie groupoids and a boutet de monvel type calculus, 2017.

[13]

V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR, 268 (1983), 285-287. 

[14]

C. Ehresmann, Catégories topologiques et catégories différentiables, In Colloque Géom. Diff. Globale (Bruxelles, 1958), Centre Belge Rech. Math., Louvain, (1959), 137–150.

[15]

B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Differential Geom., 40 (1994), 213-238.  doi: 10.4310/jdg/1214455536.

[16]

M. K. Flari and K. C. H. Mackenzie, Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.  doi: 10.1007/s11005-018-1103-y.

[17]

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.

[18]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.

[19]

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.

[20]

M. Gualtieri, Generalized complex geometry, Ann. of Math., 174 (2011), 75-123.  doi: 10.4007/annals.2011.174.1.3.

[21]

A. Haefliger, Groupoïdes d'holonomie et classifiants, Transversal Structure of Foliations (Toulouse, 1982), 116 (1984), 70-97. 

[22]

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

[23]

P. J. Higgins and K. C. H. Mackenzie, Fibrations and quotients of differentiable groupoids, J. London Math. Soc., 42 (1990), 101-110.  doi: 10.1112/jlms/s2-42.1.101.

[24]

P. J. Higgins and K. C. H. Mackenzie, Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures, Math. Proc. Cambridge Philos. Soc., 114 (1993), 471-488.  doi: 10.1017/S0305004100071760.

[25]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), 157-216.  doi: 10.1023/B:MATH.0000027508.00421.bf.

[26]

Y. Kosmann-Schwarzbach and K. C. H. Mackenzie, Differential operators and actions of Lie algebroids, In Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 213–233.

[27]

Z.-J. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  doi: 10.4310/jdg/1214459842.

[28]

K. C. H. Mackenzie, Rigid cohomology of topological groupoids, J. Austral. Math. Soc. Ser. A, 26 (1978), 277-301.  doi: 10.1017/S1446788700011794.

[29]

K. C. H. Mackenzie, Cohomology of Locally Trivial Groupoids and Lie Algebroids, PhD thesis, Monash University, 1979. doi: 10.1017/S0004972700011187.

[30]

K. C. H. Mackenzie, Infinitesimal theory of principal bundles, 50th ANZAAS Conference, Adelaide.

[31]

K. C. H. Mackenzie, Integrability obstructions for extensions of Lie algebroids, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), 29-52. 

[32] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, vol. 124 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.
[33]

K. C. H. Mackenzie, A note on Lie algebroids which arise from groupoid actions, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), 283-302. 

[34]

K. C. H. Mackenzie, Infinitesimal characterization of homogeneous bundles, Proc. Amer. Math. Soc., 103 (1988), 1271-1277.  doi: 10.1090/S0002-9939-1988-0955021-1.

[35]

K. C. H. Mackenzie, On extensions of principal bundles, Ann. Global Anal. Geom., 6 (1988), 141-163.  doi: 10.1007/BF00133036.

[36]

K. C. H. Mackenzie, Classification of principal bundles and Lie groupoids with prescribed gauge group bundle, J. Pure Appl. Algebra, 58 (1989), 181-208.  doi: 10.1016/0022-4049(89)90157-6.

[37]

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.

[38]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.

[39]

K. C. H. Mackenzie, Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids, Electron. Res. Announc. Amer. Math. Soc., 4 (1998), 74-87.  doi: 10.1090/S1079-6762-98-00050-X.

[40]

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.

[41]

K. C. H. Mackenzie, Affinoid structures and connections, In Poisson Geometry (Warsaw, 1998), Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 51 (2000), 175–186.

[42]

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

[43]

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

[44]

K. C. H. Mackenzie, A unified approach to Poisson reduction, Lett. Math. Phys., 53 (2000), 215-232.  doi: 10.1023/A:1011055510672.

[45]

K. C. H. Mackenzie, On certain canonical diffeomorphisms in symplectic and Poisson geometry, In Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Amer. Math. Soc., Providence, RI, 315 (2002), 187–198. doi: 10.1090/conm/315/05480.

[46]

K. C. H. Mackenzie, Duality and triple structures, In The Breadth of Symplectic and Poisson Geometry, Progr. Math., Birkhäuser Boston, Boston, MA, 232 (2005), 455–481. doi: 10.1007/0-8176-4419-9_15.

[47]

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.

[48]

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.

[49]

K. C. H. Mackenzie, Proving the Jacobi identity the hard way, In Geometric Methods in Physics, Trends Math., Birkhäuser/Springer, Basel, (2013), 357–366.

[50]

K. C. H. Mackenzie, From symplectic groupoids to double structures, In Geometric, Algebraic and Topological Methods for Quantum Field Theory, World Sci. Publ., Hackensack, NJ, (2017), 192–219.

[51]

K. C. H. Mackenzie and T. Mokri, Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids, Geom. Dedicata, 77 (1999), 317-330.  doi: 10.1023/A:1005180403695.

[52]

K. C. H. Mackenzie, A. Odzijewicz and A. Sliżewska, Poisson geometry related to Atiyah sequences, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), 29pp. doi: 10.3842/SIGMA.2018.005.

[53]

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.

[54]

K. C. H. Mackenzie and P. Xu, Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford Ser., 49 (1998), 59-85.  doi: 10.1093/qmathj/49.1.59.

[55]

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

[56]

W. Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal., 27 (1967/68), 1-32.  doi: 10.1007/BF00276433.

[57]

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.

[58]

M. Stienon and P. Xu, Atiyah classes and kontsevich-duflo type theorem for dg manifolds, Banach Center Publications, 123.

[59]

S. Vassout, Raconte-moi... un groupoïde de Lie, Gazette des Mathématiciens, 147 (2016), 58-61. 

[60]

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

[61]

A. Weinstein, Lagrangian mechanics and groupoids, In Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 7 (1996), 207–231.

Figure 1.  Kirill Mackenzie at home in Sheffield, 2020. (Courtesy of Miki Lazner.)
Figure 2.  Kirill Mackenzie in the first conference on Poisson Geometry, at Hangzhou (China) in 1999. (Courtesy of Yvette Kosmann-Schwarzbach.)
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