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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar
[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[56]

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

[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.  Google Scholar

[58]

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

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

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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar
[33]

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

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[56]

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

[57]

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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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