# American Institute of Mathematical Sciences

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
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##### References:
 [1] 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 [2] Víctor Manuel Jiménez Morales, Manuel De León, Marcelo Epstein. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. Journal of Geometric Mechanics, 2019, 11 (3) : 301-324. doi: 10.3934/jgm.2019017 [3] Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004 [4] Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 [5] Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213 [6] Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295 [7] K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87. [8] Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81 [9] Madeleine Jotz Lean, Kirill C. H. Mackenzie. Transitive double Lie algebroids via core diagrams. Journal of Geometric Mechanics, 2021, 13 (3) : 403-457. doi: 10.3934/jgm.2021023 [10] Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064 [11] Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469 [12] José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213 [13] Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243 [14] Gianne Derks. Book review: Geometric mechanics. Journal of Geometric Mechanics, 2009, 1 (2) : 267-270. doi: 10.3934/jgm.2009.1.267 [15] Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001 [16] Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295 [17] David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353 [18] Abdellatif Moudafi, Paul-Emile Mainge. Copositivity meets D. C. optimization. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021022 [19] Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10. [20] Lakehal Belarbi. Ricci solitons of the $\mathbb{H}^{2} \times \mathbb{R}$ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010