Quantum Riemannian Geometry Grundlehren der mathematischen Wissenschaften 355 1st Edition by Edwin J Beggs, Shahn Majid ISBN 3030302938 9783030302931

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ISBN 10: 3030302938 
ISBN 13: 9783030302931
Author: Edwin J Beggs, Shahn Majid

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. Thebook 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.

Quantum Riemannian Geometry Grundlehren der mathematischen Wissenschaften 355 1st Table of contents:

1 Differentials on an Algebra
1.1 First-Order Differentials
1.2 Differentials on Polynomial Algebras
1.3 Quantum Metrics and Laplacians
1.4 Differentials on Finite Sets
1.5 Exterior Algebra and the de Rham Complex
1.6 Exterior Algebras of Enveloping and Group Algebras
1.6.1 Enveloping Algebras
1.6.2 Group Algebras
1.7 The Exterior Algebra of a Finite Group
1.7.1 Left Covariant Exterior Algebras
1.7.2 Bicovariant Exterior Algebras
1.7.3 Finite Lie Theory and Cohomology
1.8 Application to Naive Electromagnetism on Discrete Groups
1.9 Application to Stochastic Calculus
Exercises for Chap. 1
Notes for Chap. 1
2 Hopf Algebras and Their Bicovariant Calculi
2.1 Hopf Algebras
2.2 Basic Examples of Hopf Algebras
2.3 Translation-Invariant Integrals and Differentials
2.4 Monoidal and Braided Categories
2.5 Bicovariant Differentials on Coquasitriangular Hopf Algebras
2.6 Braided Exterior Algebras
2.7 The Lie Algebra of a Quantum Group
2.7.1 Bicovariant Quantum Lie Algebras
2.7.2 Braided Lie Algebras
2.8 Bar Categories
Exercises for Chap. 2
Notes for Chap. 2
3 Vector Bundles and Connections
3.1 Finitely Generated Projective Modules
3.2 Covariant Derivatives
3.3 K-Theory and Cyclic Cohomology
3.3.1 C*-Algebras and Hilbert Spaces
3.3.2 K-Theory and Completions
3.3.3 Hochschild Homology and Cyclic Homology
3.3.4 Pairing K-Theory and Cyclic Cohomology
3.3.5 Twisted Cycles
3.4 Bimodule Covariant Derivatives
3.4.1 Bimodule Connections on Hopf Algebras
3.4.2 The Monoidal Category of Bimodule Connections
3.4.3 Conjugates of Bimodule Connections
3.5 Line Modules and Morita Theory
3.6 Exact Sequences and Abelian Categories
3.6.1 Exact Sequences, Flat and Projective Modules
3.6.2 Abelian Categories
Exercises for Chap. 3
Notes for Chap. 3
4 Curvature, Cohomology and Sheaves
4.1 Differentiating Module Maps and Curvature
4.2 Coactions on the de Rham Complex
4.3 Sheaf Cohomology
4.4 Spectral Sequences and Fibrations
4.4.1 The Spectral Sequence of a Resolution
4.4.2 The van Est Spectral Sequence
4.4.3 Fibrations and the Leray–Serre Spectral Sequence
4.5 Correspondences, Bimodules and Positive Maps
4.5.1 B-A Bimodules with Connections
4.5.2 Hilbert C*-Bimodules and Positive Maps
4.6 Relative Cohomology and Cofibrations
Exercises for Chap. 4
Notes for Chap. 4
5 Quantum Principal Bundles and Framings
5.1 Universal Calculus Quantum Principal Bundles
5.1.1 Hopf–Galois Extensions
5.1.2 Trivial Quantum Principal Bundles
5.2 Constructions of Quantum Bundles with Universal Calculus
5.2.1 Galois Field Extensions as Quantum Bundles
5.2.2 Quantum Homogeneous Bundles with Universal Calculus
5.2.3 Line Bundles and Principal Bundles
5.3 Associated Bundle Functors
5.4 Quantum Bundles with Nonuniversal Calculus
5.4.1 Quantum Principal Bundles
5.4.2 Strong Quantum Principal Bundles
5.5 Principal Bundles and Spectral Sequences
5.6 Quantum Homogeneous Spaces as Framed Quantum Spaces
5.6.1 Framed Quantum Manifolds
5.6.2 Trivially Framed Quantum Manifolds
Exercises for Chap. 5
Notes for Chap. 5
6 Vector Fields and Differential Operators
6.1 Vector Fields and Their Action
6.2 Higher Order Differential Operators
6.3 TX• as an Algebra in Z(AEA)
6.4 The Sheaf of Differential Operators DA
6.5 More Examples of Algebras of Differential Operators
6.5.1 Left-Invariant Differential Operators on Hopf Algebras
6.5.2 A Noncommutative Witt Enveloping Algebra
6.5.3 Differential Operators on M2(C)
Exercises for Chap. 6
Notes for Chap. 6
7 Quantum Complex Structures
7.1 Complex Structures and the Dolbeault Double Complex
7.2 Holomorphic Modules and Dolbeault Cohomology
7.3 Holomorphic Vector Fields
7.4 The Borel–Weil–Bott Theorem and Other Topics
7.4.1 Positive Line Bundles and the Borel–Weil–Bott Theorem
7.4.2 A Representation of a Noncommutative Complex Plane
Exercises for Chap. 7
Notes for Chap. 7
8 Quantum Riemannian Structures
8.1 Bimodule Quantum Levi-Civita Connections
8.2 More Examples of Bimodule Riemannian Geometries
8.2.1 Riemannian Geometry with Grassmann Exterior Algebra
8.2.2 Riemannian Geometry of Graphs and Finite Groups
8.2.3 The Riemannian Structure of q-Deformed Examples
8.3 Wave Operator Quantisation of C∞(M)R
8.4 Hermitian Riemannian Geometry
8.5 Geometric Realisation of Spectral Triples
8.5.1 Construction of Spectral Triples from Connections
8.5.2 Examples of Geometric Spectral Triples
8.5.3 A Dirac Operator for Endomorphism Calculi
8.6 Hermitian Metrics and Chern Connections
Exercises for Chap. 8
Notes for Chap. 8
9 Quantum Spacetime
9.1 The Quantum Spacetime Hypothesis
9.2 Bicrossproduct Models and Variable Speed of Light
9.2.1 Classical Data for the 2D Model and Planckian Bound
9.2.2 The Flat Bicrossproduct Model and Its Wave Operator
9.2.3 The Spin Model and 3D Quantum Gravity
9.3 The Quantum Black-Hole Wave Operator
9.3.1 Algebraic Methods and Polar Coordinates
9.3.2 Quantum Wave Operators on Spherically Symmetric Static Spacetimes
9.4 Curved Quantum Geometry of the 2D Bicrossproduct Model
9.4.1 Emergence of the Bicrossproduct Model Quantum Metric
9.4.2 Quantum Connections for the Bicrossproduct Model
9.5 Bertotti–Robinson Quantum Spacetimes
9.5.1 Emergence of the Bertotti–Robinson Quantum Metric
9.5.2 The Quantum Connection for the Bertotti–Robinson Model
9.6 Poisson–Riemannian Geometry and Nonassociativity
9.6.1 Semiquantisation Constructions
9.6.2 Some Solutions of the PRG Equations
9.6.3 Quantisation by Twisting

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Tags: Edwin J Beggs, Shahn Majid, Quantum Riemannian, Geometry Grundlehren