02990cam a2200325Ia 4500001001300000003000600013007000300019008004100022020001800063020001500081035002200096040002900118050002200147100002000169245006400189260002800253300002400281490011100305504005000416505035700466520155500823650002802378650001702406650002602423700001802449830006302467942001202530952010702542999001502649on1251847641OCoLCta210520s2020    sz a          000 0 eng d  a9783030302962  a3030302962  a(OCoLC)1251847641  aTULIBbengcTULIBdTULIB  aQA641b.B444 20200 aBeggs, Edwin J.10aQuantum riemannian geometry /cEdwin J. Beggs, Shahn Majid.  aCham :bSpringer,c2020  axvi, 809 p. :bill.1 aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics;vvol. 355  aIncludes bibliographical reference and index.0 aDifferentials 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.  a"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." -- Prové de l'editor. 4aGeometry, Differential. 4aGravitation. 4aMathematical physics.1 aMajid, Shahn. 0aGrundlehren der mathematischen Wissenschaften ;vvol. 355.  2lcccBK  00104070aPNLIBbPNLIBcGENd2021-06-17oQA641 .B444 2020pPNLIB21062212r2021-06-17w2021-06-17yBK  c2399d2399