September  2021, 13(3): 277-283. doi: 10.3934/jgm.2021026

Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie

School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

Reprinted with permission. Originally published as: Theodore Voronov, Book review: General theory of Lie groupoids and Lie algebroids (London Mathematical Society Lecture Note Series 213) By Kirill C. H. Mackenzie: xxxviii+501 pp., £50.00 (US$90.00) (LMS members' price £37.50 (US$67.50)), isbn 0-521-49928-3 (Cambridge University Press, Cambridge, 2005). Bull. Lond. Math. Soc. 42 (2010), no. 1,185–190. © 2010 London Mathematical Society. doi: 10.1112/blms/bdp115. Published online 5 January 2010.

Received  August 2021 Published  September 2021 Early access  September 2021

Citation: Theodore Voronov. Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie. Journal of Geometric Mechanics, 2021, 13 (3) : 277-283. doi: 10.3934/jgm.2021026
References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207.  doi: 10.1090/S0002-9947-1957-0086359-5.  Google Scholar

[2]

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

[3]

R. Brown, From groups to groupoids: A brief survey, Bull. London Math. Soc., 19 (1987), 113–134, Extended version at http://www.bangor.ac.uk/r.brown/groupoidsurvey.pdf. doi: 10.1112/blms/19.2.113.  Google Scholar

[4]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Providence, RI, 1999.  Google Scholar

[5]

A. S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys., 212 (2000), 591-611.  doi: 10.1007/s002200000229.  Google Scholar

[6]

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

[7]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[8]

C. Ehresmann, Œuvres complètes et commentées. I-1, 2. Topologie algébrique et géométrie différentielle, Cahiers Topologie Géom. Différentielle, 24 (1983). Google Scholar

[9]

J.-C. Herz, Pseudo-algèbres de Lie. I, C. R. Acad. Sci. Paris, 236 (1953), 1935-1937.   Google Scholar

[10]

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

[11]

M. V. Karasëv, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 508-538.   Google Scholar

[12] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.  Google Scholar
[13]

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

[14]

K. C. H. Mackenzie, Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids, De Gruyter, 2011 (2011). doi: 10.1515/crelle.2011.092.  Google Scholar

[15]

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

[16] I. Moerdijk and J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615450.  Google Scholar
[17]

J. Pradines, Théorie de Lie pour les groupoï des différentiables. Calcul différenetiel dans la catégorie des groupoï des infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245–A248.  Google Scholar

[18]

J. Pradines, Géométrie différentielle au-dessus d'un groupoï de, C. R. Acad. Sci. Paris Sér. A-B, 266 (1968), A1194–A1196.  Google Scholar

[19]

J. Pradines, Troisième théorème de Lie les groupoï des différentiables, C. R. Acad. Sci. Paris Sér. A-B, 267 (1968), A21–A23.  Google Scholar

[20]

T. Voronov, Q-manifolds and Mackenzie theory: An overview, preprint, arXiv: 0709.4232, [math.DG]. ESI 1952, 2007. Google Scholar

[21]

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

[22]

A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, Notices Amer. Math. Soc., 43 (1996), 744-752.   Google Scholar

[23]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅰ. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370.  doi: 10.1007/BF02097706.  Google Scholar

[24]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅱ. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395.  doi: 10.1007/BF02097707.  Google Scholar

show all references

References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207.  doi: 10.1090/S0002-9947-1957-0086359-5.  Google Scholar

[2]

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

[3]

R. Brown, From groups to groupoids: A brief survey, Bull. London Math. Soc., 19 (1987), 113–134, Extended version at http://www.bangor.ac.uk/r.brown/groupoidsurvey.pdf. doi: 10.1112/blms/19.2.113.  Google Scholar

[4]

A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, American Mathematical Society, Providence, RI, 1999.  Google Scholar

[5]

A. S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys., 212 (2000), 591-611.  doi: 10.1007/s002200000229.  Google Scholar

[6]

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

[7]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[8]

C. Ehresmann, Œuvres complètes et commentées. I-1, 2. Topologie algébrique et géométrie différentielle, Cahiers Topologie Géom. Différentielle, 24 (1983). Google Scholar

[9]

J.-C. Herz, Pseudo-algèbres de Lie. I, C. R. Acad. Sci. Paris, 236 (1953), 1935-1937.   Google Scholar

[10]

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

[11]

M. V. Karasëv, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 508-538.   Google Scholar

[12] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.  Google Scholar
[13]

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

[14]

K. C. H. Mackenzie, Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids, De Gruyter, 2011 (2011). doi: 10.1515/crelle.2011.092.  Google Scholar

[15]

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

[16] I. Moerdijk and J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615450.  Google Scholar
[17]

J. Pradines, Théorie de Lie pour les groupoï des différentiables. Calcul différenetiel dans la catégorie des groupoï des infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245–A248.  Google Scholar

[18]

J. Pradines, Géométrie différentielle au-dessus d'un groupoï de, C. R. Acad. Sci. Paris Sér. A-B, 266 (1968), A1194–A1196.  Google Scholar

[19]

J. Pradines, Troisième théorème de Lie les groupoï des différentiables, C. R. Acad. Sci. Paris Sér. A-B, 267 (1968), A21–A23.  Google Scholar

[20]

T. Voronov, Q-manifolds and Mackenzie theory: An overview, preprint, arXiv: 0709.4232, [math.DG]. ESI 1952, 2007. Google Scholar

[21]

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

[22]

A. Weinstein, Groupoids: Unifying internal and external symmetry. A tour through some examples, Notices Amer. Math. Soc., 43 (1996), 744-752.   Google Scholar

[23]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅰ. Union pseudogroups and their quantization, Comm. Math. Phys., 134 (1990), 347-370.  doi: 10.1007/BF02097706.  Google Scholar

[24]

S. Zakrzewski, Quantum and classical pseudogroups. Ⅱ. Differential and symplectic pseudogroups, Comm. Math. Phys., 134 (1990), 371-395.  doi: 10.1007/BF02097707.  Google Scholar

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