Quantum Riemannian Geometry. 1st ed. 2020

種類:

電子ブック

責任表示:

by Edwin J. Beggs, Shahn Majid

出版情報:

Cham : Springer International Publishing : Imprint: Springer, 2020

著者名:

• Beggs, Edwin J.
• Majid, Shahn.
• SpringerLink (Online service)

シリーズ名:

Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics ; 355

ISBN:

9783030302948 [3030302946]    

注記:

Differentials On An Algebra -- Hopf Algebras and Their Bicovariant Calculi -- Vector Bundles and Connections -- Curvature, Cohomology and Sheaves -- Quantum Principal Bundles and Framings -- Vector Fields and Differential Operators -- Quantum Complex Structures -- Quantum Riemannian Structures -- Quantum Spacetime -- Solutions -- References -- Index.
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 Conne

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