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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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. 

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

[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.

[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.

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

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[24]

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

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.

[2]

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

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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. 

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

[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.

[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.

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

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[24]

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

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