05087nam 22008655 450 991073948170332120200706173020.03-642-36216-810.1007/978-3-642-36216-3(CKB)2670000000360686(EBL)1206113(SSID)ssj0000878612(PQKBManifestationID)11500334(PQKBTitleCode)TC0000878612(PQKBWorkID)10836476(PQKB)10693970(DE-He213)978-3-642-36216-3(MiAaPQ)EBC6314878(MiAaPQ)EBC1206113(Au-PeEL)EBL1206113(CaPaEBR)ebr10983214(OCoLC)845259482(PPN)168329905(EXLCZ)99267000000036068620130228d2013 u| 0engur|n|---|||||txtccrClifford Algebras and Lie Theory[electronic resource] /by Eckhard Meinrenken1st ed. 2013.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,2013.1 online resource (321 p.)Ergebnisse der Mathematik und ihrer Grezgebiete. ;3. Folge, Volume 58Originally published: 2013.3-642-43669-2 3-642-36215-X Includes bibliographical references and index.Preface -- Conventions -- List of Symbols -- 1 Symmetric bilinear forms -- 2 Clifford algebras -- 3 The spin representation -- 4 Covariant and contravariant spinors -- 5 Enveloping algebras -- 6 Weil algebras -- 7 Quantum Weil algebras -- 8 Applications to reductive Lie algebras -- 9 D(g; k) as a geometric Dirac operator -- 10 The Hopf–Koszul–Samelson Theorem -- 11 The Clifford algebra of a reductive Lie algebra -- A Graded and filtered super spaces -- B Reductive Lie algebras -- C Background on Lie groups -- References -- Index.This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.Ergebnisse der Mathematik und ihrer Grezgebiete. ;3. Folge, Bd. 58.Topological groupsLie groupsAssociative ringsRings (Algebra)Mathematical physicsDifferential geometryPhysicsTopological Groups, Lie Groupshttps://scigraph.springernature.com/ontologies/product-market-codes/M11132Associative Rings and Algebrashttps://scigraph.springernature.com/ontologies/product-market-codes/M11027Mathematical Applications in the Physical Scienceshttps://scigraph.springernature.com/ontologies/product-market-codes/M13120Differential Geometryhttps://scigraph.springernature.com/ontologies/product-market-codes/M21022Mathematical Methods in Physicshttps://scigraph.springernature.com/ontologies/product-market-codes/P19013Topological groups.Lie groups.Associative rings.Rings (Algebra).Mathematical physics.Differential geometry.Physics.Topological Groups, Lie Groups.Associative Rings and Algebras.Mathematical Applications in the Physical Sciences.Differential Geometry.Mathematical Methods in Physics.512.57Meinrenken Eckhardauthttp://id.loc.gov/vocabulary/relators/aut521474MiAaPQMiAaPQMiAaPQBOOK9910739481703321Clifford algebras and Lie theory836950UNINA