A Brief Introduction to Quantum Geometry

Table of Contents

Introduction
From Geometry to Algebra: 1
Part 2
Non-commutative Generalizations
Examples
Spaces With Indistinguishable Points
Quantum Groups and Quantum Bundles
Quantum Tori and Matrix Spaces
Supermanifolds
Quantization of Symplectic Manifolds
C*-Algebraic Extensions and BDF-Theory
Literature


Literature

[BR] Brateli O, Robinson D: Operator Algebras and Quantum Statistical Mechanics, Volumes 1/2, Springer-Verlag (1979)
[BDF] Brown L G, Douglas R G, Filmore P G: Extensions of C*-algebras and K-Homology, Ann Math, 105 265-324 (1977)
[C1] Connes A: Non-commutative differential geometry, Extrait des Publications Mathematiques--IHES 62 (1986)
[C2] Connes A: Noncommutative Geometry, Academic Press (1994)
[Dx] Dixmier J: Les algebres d'operateurs dans l'espace Hilbertien, Gauthier-Villars (1957)
[DubV]  Dubois-Violette: Derivations et calcul differentiel non-commutatif, CR Acad Sci Paris 307 (1988)
[DKM1]  Dubois-Violette, R Kerner, J Madore: Noncommutative Differential Geometry of Matrix Algebras, J Math Phys 31 316-322 (1990)
[DKM2]  Dubois-Violette, R Kerner, J Madore: Noncommutative Differential Geometry and New Models of Gauge Theory, J Math Phys 31 323-335 (1990)
[D1] Durdevich M: Geometry of Quantum Principal Bundles I, Commun Math Phys 175 (3) 457-521 (1996)
[D2] Durdevich M: Geometry of Quantum Principal Bundles II, Rev Math Phys 9 (5) 531-607 (1997)
[D3] Durdevich M: Algebro-Geometric Constructions of Subquantum Theories, Doctoral Thesis, Faculty of Physics, University of Belgrade, Serbia [in Serbian :)] (1993)
[F] Fedosov B: Deformation Quantization and Index Theory, Akademie Verlag GmbH, Berlin (1996)
[K1] Kastler D: Cyclic Cohomology Within Differential Envelope, Hermann, Paris (1988)
[K2] Kastler D: Introduction to Alain Connes' Non-Commutative Differential Geometry, Karpacz Lectures (1986)
[Mad] Madore J: An Introduction to Noncommutative Differential Geometry and Its Physical Applications, Book--Second Edition (1998)
[MvN1] Murray F J, von Neumann J: On Rings of Operators, Ann Math 37 116-229 (1936)
[MvN2] Murray F J, von Neumann J: On Rings of Operators 2, Trans Amer Math Soc 41 208-248 (1937)
[MvN4] Murray F J, von Neumann J: On Rings of Operators 4, Ann Math {\bf 44} 716-808 (1943)
[P] Podles P: Quantum Spheres, Lett Math Phys 14 193-202 (1987)
[PW] Podles P, Woronowicz S L: Quantum Defformation of Lorentz Group, Commun Math Phys 130 381-431 (1990)
[SV1]Sanchez-Valenzuela O A: On Supergeometric Structures, Ph.D.Thesis, Harvard University, Cambridge (1986)
[SV2]Sanchez-Valenzuela O A: Differential-Geometric Approach to Supervector Bundles, Preprint, CIMAT, Guanajuato, Mexico (1996)
[R] Rothstein M: The Axioms of Supermanifolds and a New Structure Arising From Them, Trans Amer Math Soc 297 159-180 (1986)
[We] Wegge-Olsen N E: K-Theory and C*-Algebras, Oxford Science Publications (1993)
[W1] Woronowicz S L: Pseudospaces, Pseudogroups and Pontryagin duality, Lecture Notes in Physics 116, Proceedings of the International Conference on Mathematical Physics, Lausanne (1979)
[W2] Woronowicz S L: C*-algebras Generated by Unbounded Elements, Rev Math Phys, 7 (3) 481-521 (1995)
[W3] Woronowicz S L: Twisted SU(2) group; An example of a noncommutative differential calculus, RIMS, Kyoto University 23, 117-181 (1987)
[W4] Woronowicz S L: Compact Matrix Pseudogroups, Commun Math Phys 111, 613-665 (1987)
[W5] Woronowicz S L: Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Commun Math Phys 122, 125-170 (1989)
[W6] Woronowicz S L: Tannaka-Krein Duality for Compact Matrix Pseudogroups; Twisted SU(n) groups, Invent Math 93, 35-76 (1988)

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