Abstract
This talk reviews the basic definitions of Lie and Poisson groupoids and then proposes Lie Hopf Algebroids as a possible definition for “Quantum Groupoids” — objects which generalize quantum groups on the one hand, and have Poisson groupoids as their classical limits, on the other.
Preview
Unable to display preview. Download preview PDF.
References
M. E. Mayer: “Groupoids and Lie bigebras in gauge and string theories,” in K. Bleuler and M. Werner, editors, Proceedings of the Conference on DifferentialGeometric Methods in Physics, Como, August 1987, Reidel, Dordrecht, 1988.
M. E. Mayer: “Groupoids versus principal bundles in gauge theories,” In L. L. Chau and W. Nahm, editors, Proceedings of the Conference on Differential-Geometric Methods in Physics, Tahoe City, July 1989, Plenum Press, New York, 1990.
A. Coste, P. Dazord, and A. Weinstein: “Groupoides symplectiques,” Publ. Dép. Math. Univ. de Lyon, 2/A, 1–62 (1987.
A. Weinstein: J. Math. Soc. Japan, 40, 705–727 (1988).
K. Mikami and A. Weinstein: Publ. RIMS, Kyoto Univ., 24, 121–140 (1988).
A. Weinstein: “Affine Poisson structures.” Center for Pure and Applied Math, UC Berkeley, Preprint PAM-489:1–27, February 1990; Cal-Tech Talk, February, 1990.
A. Lichnerowicz: “Quantum mechanics and deformations of geometrical dynamics,” in A. O. Barut, editor, Quantum Theory, Groups, Fields, and Particles 3–83, pages 3–82, Reidel, Dordrecht, 1983.
A. Lichnerowicz: “Applications of the deformation of algebraic structures to geometry and mathematical physics,” in M. Hazewinkel and M. Gerstenhaber, editors, Deformation Theory of Algebras and Structures and Applications, pages 855–896, Kluwer, Dordrecht, 1989.
K. Mackenzie: “Lie Groupoids and Lie Algebroids in Differential Geometry.” Vol. 124 of London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge, 1987; See the review by Kumpera, Bull. AMS 1988.
M. F. Atiyah: Trans. Amer. Math. Soc., 85, 181–207 (1957). H. Nickerson: Trans. Amer. Math. Soc., 99, 509–539 (1961).
J. Pradines, C. R. Acad. Sc., Paris, Ser. A 264, 245–248 (1967).
V. G. Drinfel'd: Sov. Math. Dokl., 27, 68–71 (1983). Also: International Congress of Mathematicians, Proceedings Berkeley, 1986.
H. Abelson and G. J. Sussman: “The Structure and Interpretation of Computer Programs,” MIT Press/McGraw-Hill, Cambridge MA, 1985, p. 75.
D. C. Ravenel: “Complex Cobordism and Stable Homotopy Groups of Spheres,” Academic Press, 1986, Appendix 1.1.
D. Kastler, M. Mekhbout, and K. H. Rehren: “Introduction to the Algebraic Theory of Superselection Sectors,” Luminy Preprint, 1990.
M. E. Mayer: “Automorphisms of C*-Algebaas, Fell Bundles, W*-Bigebras, and the Description of Internal Symmetries in Algebraic Quantum Theory,” Acta Phys. Austriaca Suppl. VIII, 177–226 (1971). “The Uses of Group-Theoretical Duality Theorems in Quantum Theory,” in Proceedings of the Conference on DifferentialGeometric Methods in Physics, K. Bleuler and A. Reetz, Eds., Bonn 1973, pp. 254–275.
M. E. Mayer: “Differentiable Cross Sections in Banach-*-Algebraic Bundles,” in Cargèse Lectures in Physics, Vol. 4, D. Kastler, Ed. Gordon and Breach, New York 1970.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Mayer, M.E. (1991). From poisson groupoids to quantum groupoids and back. In: Bartocci, C., Bruzzo, U., Cianci, R. (eds) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics, vol 375. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53763-5_53
Download citation
DOI: https://doi.org/10.1007/3-540-53763-5_53
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53763-2
Online ISBN: 978-3-540-47090-8
eBook Packages: Springer Book Archive