LEADER 04761nam 22007095 450 001 996418260003316 005 20200702114110.0 010 $a3-030-30294-6 024 7 $a10.1007/978-3-030-30294-8 035 $a(CKB)4100000010121958 035 $a(DE-He213)978-3-030-30294-8 035 $a(MiAaPQ)EBC6033325 035 $a(PPN)242845266 035 $a(EXLCZ)994100000010121958 100 $a20200131d2020 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aQuantum Riemannian Geometry$b[electronic resource] /$fby Edwin J. Beggs, Shahn Majid 205 $a1st ed. 2020. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020. 215 $a1 online resource (XVI, 809 p. 124 illus., 8 illus. in color.) 225 1 $aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,$x0072-7830 ;$v355 311 $a3-030-30293-8 320 $aIncludes bibliographical references and index. 327 $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. 330 $aThis book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up? one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita? bimodule connection, geometric Laplacians and, in some cases, Dirac operators.The book also covers elements of Connes? approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers. 410 0$aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,$x0072-7830 ;$v355 606 $aMathematical physics 606 $aGravitation 606 $aDifferential geometry 606 $aAssociative rings 606 $aRings (Algebra) 606 $aGroup theory 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 606 $aClassical and Quantum Gravitation, Relativity Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/P19070 606 $aDifferential Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21022 606 $aAssociative Rings and Algebras$3https://scigraph.springernature.com/ontologies/product-market-codes/M11027 606 $aGroup Theory and Generalizations$3https://scigraph.springernature.com/ontologies/product-market-codes/M11078 615 0$aMathematical physics. 615 0$aGravitation. 615 0$aDifferential geometry. 615 0$aAssociative rings. 615 0$aRings (Algebra). 615 0$aGroup theory. 615 14$aMathematical Physics. 615 24$aClassical and Quantum Gravitation, Relativity Theory. 615 24$aDifferential Geometry. 615 24$aAssociative Rings and Algebras. 615 24$aGroup Theory and Generalizations. 676 $a516.373 700 $aBeggs$b Edwin J$4aut$4http://id.loc.gov/vocabulary/relators/aut$01015114 702 $aMajid$b Shahn$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996418260003316 996 $aQuantum Riemannian Geometry$92368789 997 $aUNISA