A very incomplete list of papers on groupoids

A very incomplete list of papers and books on groupoids (or that mention groupoids) arranged chronologically (by publication date)

1991

Dazord, P., and Weinstein, Alan (editors), Symplectic geometry, groupoids, and integrable systems/Seminaire sudrhodanien de geometrie a Berkeley, Springer-Verlag, New York, 1991.

Doran, Robert S. (editor), Selfadjoint and nonselfadjoint operator algebras and operator theory: proceedings of the CBMS regional conference on Coordinates in Operator Algebras: Groupoids and Categories, Their Representations and Applications held May 19-26, 1990 at Texas Christian University, American Mathematical Society, Providence R.I., 1991.

Groupoids are found in the papers by Muhly, Qiu, and Solel; Peters; Ramsay and Walter; and Yamanouchi; as well as problems submitted by Paterson, Shultz, and Solel.

Kock, A., Moedijk, I., Presentations of etendues, Cahiers de Topology et Geometrie Differentielle Categoriques, 32 (1991) 145-164.

This paper discusses the relationship between pseudogroups and topological groupoids.

Latch, The connection between the fundamental groupoid and a unification algorithm for syntactil algebras, Cahiers de topologie et geometrie differentielle categoriques, 32 (1991), 203-

E.S. Ljapin and A.E. Evseev, Partial Algebraic Operations, Obrazovanie Press, St. Petersburg, 1991.

In Russian. This book was later revised and republished in English in 1997 under the title The Theory of Partial Algebraic Operations.

Renault, Jean, The ideal structure of groupoid crossed product C*-algebras (with an appendix by G. Skandalis), J. Operator Theory 25 (1991), 3-36.

Upmeier, Harald, Multivariable Toeplitz Operators and Index Theory, in Mappings of Operator Algebras, Proceedings of the Japan-U.S. Joint Seminar, University of Pennsylvania, 1988, Birkhauser, Boston, 1991, 275-288.

Upmeier uses groupoids to generalize the ideal of a solvable (type I) C*-algebra to the idea of a groupoid solvable (non type I) C*-algebra. This concept is used in studying Toeplitz operators connected with non-type I C*-algebras.

Yamagami, Shigeru, On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids, in Mappings of Operator Algebras, Proceedings of the Japan-U.S. Joint Seminar, University of Pennsylvania, 1988, Birkhauser, Boston, 1991, 199-204.

With some stringent conditions on the groupoid G, Yamagami presents a concrete realization of the primitive ideal space of C*(G,A) where A is a C*-algebra over G.

Weinstein, A. and Xu, P., Extensions of symplectic groupoids and quantization, J. Reine agnew. Math. 417 (1991), 159-189.

1992

Aof, M.A., and Brown, R., The holonomy groupoid of a locally topological groupoid, Topology Appl. 47 (1992), 97-113.

Brown, Ronald, and Mackenzie, K.C.H., Determinations of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.

Lawson, M.V., The geometric theory of inverse semigroups II: E-unitary covers of inverse semigroups, Journal of Pure and Applied Algebra, 83 (1992) 121-139.

This paper applies ordered groupoids to inverse semigroups.

Maltsiniotis, G., Groupoides quantiques, C. R. Acad. Sci. Paris 314 (1992), 249-252.

Muhly, P.S. and Williams, D.P., Continuous trace groupoid C*-algebras, Math. Scand. 70 (1992), 127-145.

Solel, B., Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras, Ergodic theory and dynamical systems, 12 (1992), 341-

Vainerman, L., A note on quantum groupoids, C.R. Acad. Sci. Paris Ser. I Math. 315 (1992), 1125-1130.

Weinstein, Alan, and Xu, P., Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148 (1992), 309-343.

The authors introduce a groupoid-theoretical Yang-Baxter equation.

Xu, P., Symplectic groupoids of reduced Poisson spaces, C.R. Acad. Sci. Paris Ser. I Math. 314 (1992), 457-461.

Yamanouchi, Takehiko, Crossed products by groupoid actions and their smooth flows of weights, Publications of the Research Institute for Mathematical Sciences, 28 (1992), 535-

Yamanouchi, Takehiko, Dual weights on crossed products by groupoid actions, Publications of the Research Institute for Mathematical Sciences, 28 (1992), 653-

1993

Joyal, A., Tierney, M., Classifying spaces for sheaves of simplicial groupoids, Journal of pure and applied algebra, 89 (1993), 135-

Lawson, M.V., Congruences on ordered groupoids, Semigroup Forum 47 (1993), 150-167.

Lawson, M.V., An equivalence theorem for inverse semigroups, Semigroup Forum 47 (1993), 7-14.

This paper applies ordered groupoids to inverse semigroups.

Yamanouchi, Takehiko, Duality for actions and coactions of finite groupoids on von Neumann algebras, Journal of Algebra, 162 (1993), 436-

Yamanouchi, Takehiko, Duality for actions and coactions of measured groupoids of von Neumann algebras, American Mathematical Society, Providence, R.I., 1993.

1994

Brylinski, J-L. and Nistor, V., Cyclic cohomology of etale groupoids, K-theory 8 (1994), 341-365.

Connes, A. Noncommutative geometry, Academic Press, San Diego, 1994.

This book touches on groupoids briefly in Connes' exhaustive treatment of noncommutative geometry. Connes includes a section on the tangent groupoid of a manifold. Connes also claims that Heisenberg used an equivalence relation groupoid (and it's associated matrix algebra) in his development of quantum mechanics, although Heisenberg did not recognize the object he was studying as being known to mathematicians. Connes uses this example as a way to remove the following prejudice that he claims mathematicians have: "It is fashionable among mathematicians to despise groupoids and to consider that only groups have an authentic mathematical status, probably because of the pejorative suffix oid."

Mackenzie, K.C.H., and Xu, P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 18 (1994), 415-452.

Muhly, Paul. S., and Solel, B., Dilations and Commutant Lifting for Subalgebras of Groupoid C*-Algebras, International Journal of Mathematics, 5 (1994), 87-

Nica, A., On a groupoid construction for actions of certain inverse semigroups, International J. Math. 5 (1994), 349-372.

Paterson, Alan L.T., Inverse semigroups, groupoids and a problem of J. Renault in Algebraic Methods in Operator Theory (eds. Raul E. Curto and Palle E.T. Jorgensen), Birkhauser, Boston, 1994.

The author shows that the answer to the problem "If S is a unital inverse semigroup, does there exist an r-discrete groupoid G with S 'determining' G as an inverse subsemigroup of the ample semigroup of G? What kind of uniqueness can we expect?" is no. This problem was raised in J. Renault's book A groupoid approach to C*-algebras.

Renault, Jean, Multiplicateurs de Fourier et fonctions de Littlewood pour les groupoides r-discrets, C.R. Acad Sci. Paris Ser. I Math. 319 (1994), 15-19.

Vaismann, I., Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, Vol. 118, Birkhauser, Boston, 1994.

This book written for readers with a familiarity with the general geometry of differentiable manifolds, contains one chapter on the "Realization of Poisson Manifolds by Symplectic Groupoids."

Yamanouchi, Takehiko, Duality for Generalized Kac Algebras and a Characterization of Finite Groupoid Algebras, Journal of algebra, 163 (1994), 5-50.

1995

Brown, Ronald, and Mucuk, O., The monodromy groupoid of a Lie groupoid, Cahiers Topologie Geom. Differentielle Categ. 36 (1995), 345-369.

Bryant, Darryn E., Homomorphisms of groupoids related to graph decompositions, The Journal of Combinatorial Mathematics and Combinatorial Computing, 17 (1995) 129-

Deaconu, Valentin, Groupoids associated with endomorphisms, Trans. Amer. Math. Soc. 347 (1995), 1779-1786.

Hilgert, J., Neeb, K.-H., Groupoid C*-algebras of order compactified symmetric spaces, Japanese Journal of Mathematics, 21 (1995), 117-

Kellendonk, J., Noncommutative geometry of tilings and gap labeling, Reviews in Mathematical Physics 7 (1997) 115-157.

In this paper is a description of a method for constructing a topological groupoid from an inverse semigroup.

Muhly, P.S. and Williams, D.P., Groupoid cohomology and the Dixmier-Douady class, Proc. London Math. Soc. 71 (1995), 109-134.

Ramsay, Arlan and Walter, Martin, Noncommutative Harmonic Analysis on Groupoids in Topics in Operator Theory, Operator Algebras and Applications: 15th International Conference on Operator Theory, Timisoara (Romania), June 6-10, 1994, Bucharest, Institute of Mathematics of the Romanian Academy, 1995.

This paper contains the outline and main ideas of the paper published in 1997 in the Journal of Functional Analysis.

Xiaoman, Chen, Qingxiang, Xu, and Shengzhi, Xu, Irrational Rotation C*-Algebra for Groupoid C*-Algebra, Chinese annals of mathematics, Ser. B., 16 (1995), 445-

Xu, P., On Poisson groupoids, International J. Math. 6 (1995), 101-124.

1996

Bohm, G., Szlachanyi, K., A coassociative C*-quantum group with nonintegral dimensions, Lett. in Math. Phys., 35 (1996), 437-456.

Introduces the concept of weak Hopf algebras which are finite quantum groupoids.

Brown, Ronald, Homotopy theory and change of base of groupoids and multiple groupoids, The European Colloquium of Category Theory (Tours, 1994), Appl. Categ. Structures 4 (1996), 175-193.

Brown, Ronald, and Mucuk, O., Foliations, locally Lie groupoids and holonomy, Cahiers Topologie Geom. Differentielle Categ. 37 (1996), 61-71.

Dicks, Warren and Ventura, Enric, The Group Fixed by a Family of Injective Endomorphisms of a Free Group, in Contemporary Mathematics, 195, Providence, American Mathematical Society, 1996.

Groupoids are used to reprove many of the theorems dealing with automorphisms and endomorphisms of a finitely generated free group.

Haataja, S., Margolis, S.W., and Meakin, J., Bass-Serre theory for groupoids and the structure of full semigroup amalgams, Journal of Algebra, 183 (1996), 38-54.

This paper applies ordered groupoids to inverse semigroups.

Joyal, A. Tierney, M., On the homotopy theory of sheaves of simplicial groupoids, Mathematical proceedings of the Cambridge Phil. Soc, 120 (1996), 263-

Lawson, M.V., A class of actions of inverse semigroups, Journal of Algebra 179 (1996), 425-460.

This paper applies ordered groupoids to inverse semigroups.

Liu, Z-J. and Xu, P., Exact Lie bialgebroids and Poisson groupoids, Geom.. and Funct. Anal. 6 (1996) 138-145.

Lu, J.-H., Hopf Algebroids and Quantum Groupoids, International Journal of Mathematics, 7 (1996) 47-

Muhly, P.S., Renault, J.N., and Williams, D.P., Continuous-trace groupoid C*-algebras III., Trans. Amer. Math. Soc. 348 (1996), 3621-3641.

Muhly, Paul S., Solel, Baruch, Representations of triangular subalgebras of groupoid C*-algebras, Journal of the Australian Mathematical Society, Ser. A., 61 (1996), 289-

Weinstein, Alan, Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), 207-231, Fields Inst. Commun. 7, American Mathematical Society, R.I., 1996.

Weinstein, Alan, "Groupoids: Unifying Internal and External Symmetry, A Tour through some examples," Notices of the AMS, July 1996, 744 - 751.

This paper is an introduction to groupoids aimed at general mathematicians. It motivates the definition and some of the properties through looking at various symmetries that are not explained fully by groups. In addition to some (brief) history, this paper also talks about Lie groupoids and Lie algebroids.

1997

Dazord, P., Groupoide d'holonomie et geometrie globale, C.R. Acad. Sci. Paris Ser. I Math. 324 (1997), 77-80.

Guruprasad, J. Hebschmann, Jeffrey, L., and Weinstein, A., Group systems, groupoids and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997) 377-412.

Heller, M. and Sasin, W., Groupoid approach to noncommutative quantization of gravity, J. Math. Phys. 38 (1997), 5840-5853.

Keel, S., and Mori, S., Quotients by groupoids, Ann. of Math. 145 (1997) 193-213.

Kellendonk, J., Topological equivalence of tilings, Journal of Mathematical Physics 38 (1997), 1823-1842.

In this paper is a description of a method for constructing a topological groupoid from an inverse semigroup.

Kumijian, A., Pask, D., Raeburn, I., and Renault, J., Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505-541.

Ljapin, E.S. and Evseev, A.E., The Theory of Partial Algebraic Operations, Kluwer Academic Publishers, The Netherlands, 1997.

This book is a completely revised, enlarged, and updated translation of the original Russian Work Partial Algebraic Operations by the same authors in 1991.

Moerdijk, I. On the weak homotopy type of etale groupoids, Integrable systems and foliations (Montpellier, 1995), 147-156, Progress in Mathematics, Vol. 145, Birkhauser Boston, Boston, MA 1997.

Moerdijk, I., and Pronk, D.A., Orbifolds, sheaves and groupoids, K-theory, 12 (1997), 3-21.

Monthubert, B, and Pierrot, F., Indice analytique et groupoides de Lie, C.R. Acad. Sci. Paris, Ser. I, 325 (1997), 193-198.

Ramsay, Arlan, Lacunary sections for locally compact groupoids, Ergodic Theory Dynam. Systems 17 (1997), 933-940.

Ramsay, Arlan, and Walter, Martin E., Fourier-Stieltjes algebras of locally compact groupoids, J. Funct. Anal. 148 (1997), 314-367.

Renault, Jean, The Fourier algebra of a measured groupoid and its multipliers, J. Funct. Anal. 145 (1997), 455-490.

Sheu, A.J.L., Groupoid and compact quantum groups in Quantum groups and quantum spaces, Warsaw, 1995, 41-50, Banach Center Publ. 40 (1997).

Sheu, A.J.L., Compact quantum groups and groupoid C*-algebra, J. Funct. Anal. 144 (1997), 371-393.

Weinstein, Alan, Tangential deformation quantization and polarized symplectic groupoids in Deformation and symplectic geometry (Ascona, 1996), 301-314., Math. Phys. Stud. 20 Kluwer Acad. Publ., Dordrect, 1997.

Xu, P., Deformation Quantization and Quantum Groupoids, RIMS, Kyoto University, 1997.

Xu, P., Flux homomorphism on symplectic groupoids, Math. Z. 226 (1997), 575-597.

1998

Anantharaman-Delaroche, C., and Renault, Jean, Amenable Groupoids, with an appendix by E. Germain, Laboratoire de Mathematiques, Applications, et Physique Mathematique d'Orleans, 1998.

Kumjian, Alexander, Fell bundles over groupoids, Proc. Amer. Math. Soc. 126 (1998), 1115-1125.

Kumjian, Alexander, Muhly, Paul S., Renault, Jean N., and Williams, Dana P., The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998), 901-954.

Lawson, M.V., Inverse semigroups: The Theory of Partial Symmetries, World Scientific, Singapore, 1998.

The author states that one aim of the books is to show that "inverse semigroups had the potential to act as a unifying concept for applications of partial symmetries." He "incorporates Ehresmann's theory of ordered groupoids into mainstream inverse semigroup theory."

Monthubert, Bertrand, Groupoides et calcul pseudo-differentiel sur les varietes a coins, Ph.D. thesis, Universite Paris 7, 1998.

Monthubert, Bertrand, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc., 127 (1999), 2871-2881.

Pastijn, F., Inverse semigroups, groupoids, and amalgamation in Semigroups, (eds K.P. Shum, Y.Q. Guo, M. Ito, and Y. Fong), Springer-Verlag, New York, 1998.

The author gives a new proof using groupoids of the fact that every amalgam of an inverse semigroup is strongly embeddable in an inverse semigroup.

Ping Xu, Quantum groupoids and deformation quantization, C.R. Acad. Sci., Paris, Ser. I, Math. 326 (1998), 289-294.

Rajan, A.R., Ordered groupoids and normal categories in Semigroups, (eds K.P. Shum, Y.Q. Guo, M. Ito, and Y. Fong), Springer-Verlag, New York, 1998.

The author describes ordered groupoids as a particular direct sum of a groupoid and a preorder.

Sallam, El-Sayed Kamel Morsy, Cohomology of groupoid structure, Ph.D. Thesis, University of Colorado 1998.

1999

Balog, J., Feher, L., and Palla, L., The chiral WZNW phase space and its Poisson-Lie groupoids, Physics Letters B, 463 (1999), 83-92.

Bohm, G., Nill, F., and Szlachanyi, K., Weak Hopf algebras I: Integral theory and C*-structure, J. Algebra, 221 (1999), 385-438.

Introduces the concept of weak Hopf algebras which are finite quantum groupoids.

Cannas da Silva, Ana, and Weinstein, Alan, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, 10, American Mathematical Society, Rhode Island, 1999.

This book is based on notes on a topics course that Alan Weinstein taught at Berkeley in the Spring of 1997. There is a section on groupoids and their applications to noncommutative geometry.

Carinena, J.F., Clemente-Gallardo, J., Follana, E., Gracia-Bonda, J.M., Rivero, A., Varilly, J.C., Connes' tangent groupoid and strict quantization, J. Geom. Phys. 32 (1999), 79-96.

Crainic, M., Cyclic cohomology of etale groupoids: The general case, K-theory, 17 (1999), 319-365.

Landsman, N.P., Lie groupoid C*-algebras and Weyl quantization, Comm. Math. Phys., 206 (1999), 367-381.

Le Gall, Pierre-Yves, Theorie de Kasparov equivariante et groupoides I, K-theory, 16 (1999), 361-390.

MacKenzie, K.C.H. On Symplectic Double Groupoids and the Duality of Poisson Groupoids, International Journal of Mathematics, 10 (1999), 435-

MacKenzie, K.C.H. and Mokri, T., Locally vacant double Lie groupoids and the integration of matched pairs of Lie algebroids, Geometriae dedicata, 77 (1999), 317-

Monthubert, B., Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc. 12 (1999), 2871-2881.

Nistor, V., Weinstein, A., and Xu, P., Pseudodifferential operators on differential groupoids, Pacific Journal Math. 189 (1999), 117-152.

Oty, Karla J., Fourier-Stieltjes algebras of r-discrete groupoids, Journal of Operator Theory, 41 (1999), 175-

Paterson, Alan L.T., Groupoids, inverse semigroups, and their operator algebras, Birkhauser, Boston, 1999.

The main objective, as the title suggests, is to explore connections between the three concepts of groupoids, inverse semigroups and their operator algebras with a balanced approach between the details and the main ideas. From the introduction, "The author has tried to preserve this balance and hopes that the approach adopted in the book will contribute to an appreciation of the intrinsic beauty and importance of groupoids, and help in overcoming a not uncommon psychological aversion to the concept that he himself initially experienced."

Quigg, John and Sieben, Nandor, C*-actions of r-discrete groupoids and inverse semi-groups, Journal of the Australian Mathematical Society, Series A, Pure Mathematics and Statistics, 66 (1999), 143-

Ramazan, B., Limite classique de C*-algebres de groupoides de Lie, C.R. Acad. Sci. Paris I, 329 (1999), 603-606.

Stadler, Macho, and O'Uchi, M., Correspondence of groupoid C*-algebras, Journal of Operator Theory, 42 (1999), 103-

2000

Brown, Ronald, Icen, I., and Mucuk, O., Holonomy and monodromy groupoids, Banach Center Publications, 54 (2000), 9-20.

Crainic, M., and Moerdijk, I., A homology theory for etale groupoids, J. Reine Angew. Math. 521 (2000), 25-46.

Enock, Michel, Inclusions of Von Neumann Algebras and Quantum Groupoids II, Journal of Functional Analysis, 178 (2000), 156-

Landsman, N.P., The Muhly-Renault-Williams Theorem for Lie Groupoids and its Classical Counterpart, Letters in Mathematical Physics, 54 (2000), 43-59.

Lu, Jiang-Hua, Yan, Min, and Zhu, Yongchang, Quasi-triangular structures on Hopf algebras with positive bases in New Trends in Hopf Algebra Theory: Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Sierras de Cordoba, Argentina, August 9-13, 1999, Contemporary Mathematics, 267 (2000), 339-356.

The authors use groupoids to show that a positive quasi-triangular structure on particular Hopf algebras are set-theoretical.

Nikshych, Dmitri, A Duality Theorem for Quantum Groupoids in New Trends in Hopf Algebra Theory: Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Sierras de Cordoba, Argentina, August 9-13, 1999, Contemporary Mathematics, 267 (2000), 237-243.

Nikshych, Dmitri and Vainerman, Leonid, A characterization of depth 2 subfactors of II1 factors, Journal of Functional Analysis, 171 (2000), 278-307.

The authors show that finite index II1 subfactors of depth less than or equal to 2 can be characterized as C*-quantum groupoid smash products.

Nikshych, Dmitri, and Vainerman, Leonid, A Galois Correspondence for II1 Factors and Quantum Groupoids, Journal of Functional Analysis, 178 (2000), 113-

The authors extend the previous paper's results.

Nikshych, Dmitri and Vainerman, Leonid, Algebraic Versions of a Finite-Dimensional Quantum Groupoid in Hopf Algebras and Quantum Groups, Lecture notes in pure and applied mathematics, 209 (2000), 189-220.

A paper given at the colloquium "Hopf Algebras and Quantum Groups" held at the Free University of Brussels, this paper establishes the equivalence of three approaches of finite dimensional quantum groupoids. The three approaches are Yamanouchi's generalized Kac algebras, Bohm, Nill, and Szlachanyi's weak Kac algebras, and the algebraic version of Vallin's Hopf bimodules.

Nistor, V., Groupoids and the integration of Lie algebroids, J. Math. Soc. Japan, 52 (2000), 847-868.

Rybicki, T., On the group of lagrangian bisections of a symplectic groupoid, Banach Center Publications, 54 (2000), 235-247.

Schauenburg, Peter, Duals and Doubles of Quantum Groupoids (xR-Hopf Algebras) in New Trends in Hopf Algebra Theory: Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, La Falda, Sierras de Cordoba, Argentina, August 9-13, 1999, Contemporary Mathematics, 267 (2000), 273-299.

Suzuki, H., Central extensions of a Lie groupoid, Far East Journal of Mathematical Sciences, 2 (2000), 947-962.

Weinstein, Alan, Linearization Problems for Lie Algebroids and Lie Groupoids, Letters in Mathematical Physics, 52 (2000), 93-102.