This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points.
Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators.The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules.
A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.
Author(s): Edwin Beggs, Shahn Majid
Series: Grundlehren der mathematischen Wissenschaften 355
Publisher: Springer
Year: 2020
Language: English
Pages: 826
Preface......Page 6
Outline of Notations and Examples......Page 10
Acknowledgements......Page 12
Contents......Page 14
1 Differentials on an Algebra......Page 18
1.1 First-Order Differentials......Page 19
1.2 Differentials on Polynomial Algebras......Page 26
1.3 Quantum Metrics and Laplacians......Page 31
1.4 Differentials on Finite Sets......Page 35
1.5 Exterior Algebra and the de Rham Complex......Page 39
1.6.1 Enveloping Algebras......Page 47
1.6.2 Group Algebras......Page 56
1.7 The Exterior Algebra of a Finite Group......Page 62
1.7.1 Left Covariant Exterior Algebras......Page 63
1.7.2 Bicovariant Exterior Algebras......Page 65
1.7.3 Finite Lie Theory and Cohomology......Page 73
1.8 Application to Naive Electromagnetism on Discrete Groups......Page 82
1.9 Application to Stochastic Calculus......Page 88
Exercises for Chap. 1......Page 95
Notes for Chap. 1......Page 97
2.1 Hopf Algebras......Page 100
2.2 Basic Examples of Hopf Algebras......Page 107
2.3 Translation-Invariant Integrals and Differentials......Page 115
2.4 Monoidal and Braided Categories......Page 134
2.5 Bicovariant Differentials on Coquasitriangular Hopf Algebras......Page 148
2.6 Braided Exterior Algebras......Page 160
2.7 The Lie Algebra of a Quantum Group......Page 184
2.7.1 Bicovariant Quantum Lie Algebras......Page 187
2.7.2 Braided Lie Algebras......Page 191
2.8 Bar Categories......Page 206
Exercises for Chap. 2......Page 218
Notes for Chap. 2......Page 220
3 Vector Bundles and Connections......Page 224
3.1 Finitely Generated Projective Modules......Page 225
3.2 Covariant Derivatives......Page 235
3.3.1 C*-Algebras and Hilbert Spaces......Page 246
3.3.2 K-Theory and Completions......Page 250
3.3.3 Hochschild Homology and Cyclic Homology......Page 253
3.3.4 Pairing K-Theory and Cyclic Cohomology......Page 257
3.3.5 Twisted Cycles......Page 261
3.4 Bimodule Covariant Derivatives......Page 265
3.4.1 Bimodule Connections on Hopf Algebras......Page 268
3.4.2 The Monoidal Category of Bimodule Connections......Page 275
3.4.3 Conjugates of Bimodule Connections......Page 280
3.5 Line Modules and Morita Theory......Page 286
3.6.1 Exact Sequences, Flat and Projective Modules......Page 293
3.6.2 Abelian Categories......Page 299
Exercises for Chap. 3......Page 304
Notes for Chap. 3......Page 306
4 Curvature, Cohomology and Sheaves......Page 310
4.1 Differentiating Module Maps and Curvature......Page 312
4.2 Coactions on the de Rham Complex......Page 335
4.3 Sheaf Cohomology......Page 349
4.4 Spectral Sequences and Fibrations......Page 357
4.4.1 The Spectral Sequence of a Resolution......Page 360
4.4.2 The van Est Spectral Sequence......Page 361
4.4.3 Fibrations and the Leray–Serre Spectral Sequence......Page 365
4.5.1 B-A Bimodules with Connections......Page 376
4.5.2 Hilbert C*-Bimodules and Positive Maps......Page 381
4.6 Relative Cohomology and Cofibrations......Page 392
Exercises for Chap. 4......Page 395
Notes for Chap. 4......Page 398
5 Quantum Principal Bundles and Framings......Page 402
5.1 Universal Calculus Quantum Principal Bundles......Page 403
5.1.1 Hopf–Galois Extensions......Page 409
5.1.2 Trivial Quantum Principal Bundles......Page 415
5.2 Constructions of Quantum Bundles with Universal Calculus......Page 418
5.2.1 Galois Field Extensions as Quantum Bundles......Page 419
5.2.2 Quantum Homogeneous Bundles with Universal Calculus......Page 424
5.2.3 Line Bundles and Principal Bundles......Page 430
5.3 Associated Bundle Functors......Page 433
5.4 Quantum Bundles with Nonuniversal Calculus......Page 441
5.4.1 Quantum Principal Bundles......Page 442
5.4.2 Strong Quantum Principal Bundles......Page 459
5.5 Principal Bundles and Spectral Sequences......Page 465
5.6 Quantum Homogeneous Spaces as Framed Quantum Spaces......Page 471
5.6.1 Framed Quantum Manifolds......Page 476
5.6.2 Trivially Framed Quantum Manifolds......Page 487
Exercises for Chap. 5......Page 497
Notes for Chap. 5......Page 499
6 Vector Fields and Differential Operators......Page 502
6.1 Vector Fields and Their Action......Page 504
6.2 Higher Order Differential Operators......Page 510
6.3 TX• as an Algebra in Z(AEA)......Page 520
6.4 The Sheaf of Differential Operators DA......Page 525
6.5.1 Left-Invariant Differential Operators on Hopf Algebras......Page 532
6.5.2 A Noncommutative Witt Enveloping Algebra......Page 536
6.5.3 Differential Operators on M2(C)......Page 538
Exercises for Chap. 6......Page 541
Notes for Chap. 6......Page 542
7 Quantum Complex Structures......Page 544
7.1 Complex Structures and the Dolbeault Double Complex......Page 545
7.2 Holomorphic Modules and Dolbeault Cohomology......Page 555
7.3 Holomorphic Vector Fields......Page 565
7.4 The Borel–Weil–Bott Theorem and Other Topics......Page 569
7.4.1 Positive Line Bundles and the Borel–Weil–Bott Theorem......Page 570
7.4.2 A Representation of a Noncommutative Complex Plane......Page 575
Exercises for Chap. 7......Page 578
Notes for Chap. 7......Page 580
8 Quantum Riemannian Structures......Page 582
8.1 Bimodule Quantum Levi-Civita Connections......Page 584
8.2 More Examples of Bimodule Riemannian Geometries......Page 598
8.2.1 Riemannian Geometry with Grassmann Exterior Algebra......Page 600
8.2.2 Riemannian Geometry of Graphs and Finite Groups......Page 605
8.2.3 The Riemannian Structure of q-Deformed Examples......Page 617
8.3 Wave Operator Quantisation of C∞(M)R......Page 620
8.4 Hermitian Riemannian Geometry......Page 632
8.5 Geometric Realisation of Spectral Triples......Page 642
8.5.1 Construction of Spectral Triples from Connections......Page 643
8.5.2 Examples of Geometric Spectral Triples......Page 647
8.5.3 A Dirac Operator for Endomorphism Calculi......Page 654
8.6 Hermitian Metrics and Chern Connections......Page 658
Exercises for Chap. 8......Page 665
Notes for Chap. 8......Page 667
9 Quantum Spacetime......Page 670
9.1 The Quantum Spacetime Hypothesis......Page 671
9.2 Bicrossproduct Models and Variable Speed of Light......Page 676
9.2.1 Classical Data for the 2D Model and Planckian Bound......Page 680
9.2.2 The Flat Bicrossproduct Model and Its Wave Operator......Page 683
9.2.3 The Spin Model and 3D Quantum Gravity......Page 689
9.3.1 Algebraic Methods and Polar Coordinates......Page 696
9.3.2 Quantum Wave Operators on Spherically Symmetric Static Spacetimes......Page 704
9.4.1 Emergence of the Bicrossproduct Model Quantum Metric......Page 709
9.4.2 Quantum Connections for the Bicrossproduct Model......Page 716
9.5.1 Emergence of the Bertotti–Robinson Quantum Metric......Page 721
9.5.2 The Quantum Connection for the Bertotti–Robinson Model......Page 725
9.6 Poisson–Riemannian Geometry and Nonassociativity......Page 729
9.6.1 Semiquantisation Constructions......Page 731
9.6.2 Some Solutions of the PRG Equations......Page 737
9.6.3 Quantisation by Twisting......Page 739
Exercises for Chap. 9......Page 750
Notes for Chap. 9......Page 753
Solutions to Exercises for Chap. 1......Page 758
Solutions to Exercises for Chap. 2......Page 762
Solutions to Exercises for Chap. 3......Page 769
Solutions to Exercises for Chap. 4......Page 774
Solutions to Exercises for Chap. 5......Page 780
Solutions to Exercises for Chap. 6......Page 787
Solutions to Exercises for Chap. 7......Page 788
Solutions to Exercises for Chap. 8......Page 791
Solutions to Exercises for Chap. 9......Page 798
References......Page 808
Index......Page 820
