Quantum Riemannian Geometry

  • Edwin J. Beggs
  • Shahn Majid

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 355)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Edwin J. Beggs, Shahn Majid
    Pages 1-82
  3. Edwin J. Beggs, Shahn Majid
    Pages 83-206
  4. Edwin J. Beggs, Shahn Majid
    Pages 207-292
  5. Edwin J. Beggs, Shahn Majid
    Pages 293-384
  6. Edwin J. Beggs, Shahn Majid
    Pages 385-484
  7. Edwin J. Beggs, Shahn Majid
    Pages 485-526
  8. Edwin J. Beggs, Shahn Majid
    Pages 527-564
  9. Edwin J. Beggs, Shahn Majid
    Pages 565-652
  10. Edwin J. Beggs, Shahn Majid
    Pages 653-740
  11. Back Matter
    Pages 741-809

About this book


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.


noncommutative geometry quantum groups Hopf algebra differential graded algebra quantum Levi-Civita bimodule connection quantum spacetime quantum gravity discrete geometry of graphs braided Lie algebra noncommutative D-modules holomorphic structure Poisson-Riemannian geometry cochain twist

Authors and affiliations

  • Edwin J. Beggs
    • 1
  • Shahn Majid
    • 2
  1. 1.Department of MathematicsSwansea UniversitySwanseaUK
  2. 2.School of Mathematical SciencesQueen Mary University of LondonLondonUK

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-30293-1
  • Online ISBN 978-3-030-30294-8
  • Series Print ISSN 0072-7830
  • Series Online ISSN 2196-9701
  • Buy this book on publisher's site