Publications

K. C. H. Mackenzie

General Theory of Lie Groupoids and Lie Algebroids,
Cambridge University Press, 2005, xxxviii + 501 pages.

(with Alfonso Gracia-Saz) Duality functors for triple vector bundles, 27 pages
arXiv:0901.0203 To appear in Letters in Mathematical Physics, 2009.
Duality and triple structures
pages 455 to 481 of The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein, Birkhauser 2005, edited by J. E. Marsden and T. S. Ratiu. math.DG/0406267

The main sequence of papers on doubles, in reverse logical order, is:

``Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids'', 34 pages. math.DG/0611799 (This is a complete rewrite of preprint math.DG/0011212.)
On symplectic double groupoids and the duality of Poisson groupoids
Internat. J. Math. 10 (1999), 435--456. math.DG/9808005
Double Lie algebroids and second-order geometry, II
Adv. Math. 154 (2000), 46--75. dg-ga/9712013
Double Lie algebroids and second-order geometry, I.
Adv. Math., 94(2):180--239, 1992. pdf file
(with R. Brown.) Determination of a double Lie groupoid by its core diagram.
J. Pure Appl. Algebra, 80(3):237--272, 1992. pdf file

An overview with background is in:

Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74--87. pdf

An introduction, written for people approaching from the Poisson/symplectic point of view, is in:

On certain canonical diffeomorphisms in symplectic and Poisson geometry
Quantization, Poisson brackets and beyond, ed. Th. Voronov, Contemporary Mathematics, 315, 2002, 187 - 198. math.DG/0210384

Other papers on aspects of doubles:

Affinoid structures and connections
Poisson Geometry, Banach Center Publications Volume 51 (2000), 175--186. pdf file
(with T. Mokri) Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids
Geom. Dedicata 77 (1999), 317--330.

A unified approach to Poisson reduction
Lett. Math. Phys. 53(3) (2000), 215--232. pdf file
(with P. J. Higgins) Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures.
Math. Proc. Cambridge Philos. Soc. 114:471-488. 1993.
If you want a good laugh, you could read the MR review of this paper. (But you surely have better things to do.)
If you do read it, note that none of the passages in quotation marks in the review actually occur in the paper.
MR refused Higgins and myself the right of reply to this review and I therefore ceased to review for MR.

(with Ping Xu) Integration of Lie bialgebroids
Topology 39 (2000), 445--467. dg-ga/9712012
(with Ping Xu.) Classical lifting processes and multiplicative vector fields.
Quarterly J. Math. Oxford (2), 49 (1998), 59--85. dg-ga/9710030
(with Ping Xu.) Lie bialgebroids and Poisson groupoids.
Duke Math. J., 73(2):415--452, 1994.

(with Y. Kosmann-Schwarzbach) Differential operators and actions of Lie algebroids
Quantization, Poisson brackets and beyond, ed. T. Voronov, Contemporary Mathematics, 315, 2002, 213 - 233. math.DG/0209337
Lie algebroids and Lie pseudoalgebras.
Bull. London Math. Soc., 27(2):97--147, 1995.
(with P. J. Higgins) Algebraic constructions in the category of Lie algebroids
J. Algebra, 129:194--230, 1990.
(with P. J. Higgins) Fibrations and quotients of differentiable groupoids
J. London Math. Soc.(2) 42:101--110, 1990.
A note on Lie algebroids which arise from groupoid actions.
Cahiers Topologie Geom. Differentielle Categ. 28:283-302. 1987.

Classification of principal bundles and Lie groupoids with prescribed gauge group bundle.
J. Pure Appl. Algebra 58:181--208, 1989.
On extensions of principal bundles.
Ann. Global Anal. Geom. 6:141-163, 1988.
Integrability obstructions for extensions of Lie algebroids.
Cahiers Topologie Geom. Differentielle Categ. 28:29-52, 1987.

(as K. A. Mackenzie) Rigid cohomology of topological groupoids
J. Austral. Math. Soc. Ser. A. 26:277-301, 1978.
`Rigid cohomology' is what became known shortly afterwards as continuous cohomology. The name `rigid' was chosen to reflect the fact that the topology of the extensions which the theory classifies is (fibrewise) the direct product of the topologies of the quotient and kernel.

Lie Groupoids and Lie Algebroids in Differential Geometry,
Cambridge University Press, 1987, xvi + 327 pages.
With the exception of the material on topological groupoids, most of this book has been incorporated in revised form into my 2005 book.
The 1987 book is still of interest for reasons of priority : none of the original material had been published (or offered for publication) elsewhere.

October 15, 2009