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Author Meinrenken, Eckhard, author.
Title Clifford algebras and lie theory / Eckhard Meinrenken.
Published Heidelberg : Springer, [2013]
Book Cover
LOCATION CALL # STATUS
 Floor4  QA199 .M51 2013    IN LIBRARY
  
Description xx, 321 pages ; 25 cm.
Content Type text
Format volume
Series Ergebnisse der Mathematik und ihrer Grenzgebiete : a series of modern surveys in mathematics ; 3. Folge, volume 58
Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 58.
Bibliography Includes bibliographical references (pages 311-315) and index.
Contents Convention -- List of Symbols -- Symmetrie bilinear forms -- Quadratic vector Spaces -- Isotropie subspaces -- Split bilinear forms -- E. Cartan-Dieudonne* Theorem -- Witt's Theorem -- Orthogonal groups for K = R, C -- Lagrangian Grassmannians -- Clifford algebras -- Exterior algebras -- Definition -- Universal property, functoriality -- Derivations -- Transposition -- Duality pairings -- Clifford algebras -- Definition and first properties -- Universal property, functoriality -- The Clifford algebras Cl(n, m) -- The Clifford algebras Cl(n) -- Symbol map and quantization map -- Transposition -- Chirality dement -- The trace and the super-trace -- Lie derivatives and contracti̧ons -- The Lie algebra q(̂²(V)) -- A formula for the Clifford product -- The Clifford algebra as a quantization -- Differential Operators -- Graded Poisson algebras -- Graded super Poisson algebras -- Poisson structures on ̂(V) -- The spin representation -- The Clifford group and the spin group -- The Clifford group -- ThegroupsPin(V) andSpin(V) -- Clifford modules -- Basic constructions -- The spinor module Sf -- The dual spinor module SF -- Irreducibility of the spinor module -- Abstract spinor modules -- Pure spinors -- The canonical bilinear pairing on spinors -- The character x : ... -- Cartan's triality principle -- The Clifford algebra C/(V) -- The Clifford algebra C/(V) -- Thegroups Spinc(V) and Pinc(V) -- Spinor modules over C/(V) -- Classification of irreducible C/(V)-modules -- Spin representation -- Applications to compact Lie groups -- Covariant and contravariant spinors -- Pull-backs and push-forwards of spinors -- Factorizations -- The Lie algebra ... -- The group SO... -- The group Spin... -- The quantization map revisited -- The symbol map in terms of the spinor module -- The symbol of elements in the spin group -- Another factorization -- The symbol of elements exp... -- Clifford exponentials versus exterior algebra exponentials -- The symbol of elements exp... -- The function A... -- Volume forms on conjugacy classes -- Enveloping algebras -- The universal enveloping algebra -- Construction -- Universal property -- Augmentation map, anti-automorphism -- Derivations -- Modules overt U (g) -- Unitary representations -- Graded or filtered Lie algebras and super Lie algebras -- Further remarks -- The Poincaré-Birkhoff-Witt Theorem -- U (g) as left-invariant differential Operators -- The enveloping algebra as a Hopf algebra -- Hopf algebras -- Hopf algebra structure on S(E) -- Hopf algebra structure on U (g) -- Primitive elements -- Coderivations -- Coderivations of S(E) -- Petracci's proof of the Poincaré-Birkhoff-Witt Theorem -- A g-representation by coderivations -- The formal vector fields ... -- Proof of Petracci's Theorem -- The center of the enveloping algebra -- Weil algebras -- Differential Spaces -- Symmetrie and tensor algebra over differential Spaces -- Homotopies -- Koszul algebras -- Symmetrization -- g-differential Spaces -- The g-differential algebra ĝ* -- g-homotopies -- The Weil algebra -- Chern-Weil homomorphisms -- The non-commutative Weil algebra ̃Wg -- Equivariant cohomology of g-differential Spaces -- Transgression in the Weil algebra -- Quantum Weil algebras -- The g-differential algebra C1(g) -- The quantum Weil algebra -- Poisson structure on the Weil algebra -- Definition of the quantum Weil algebra -- The eubie Dirac Operator -- W(g) as a level 1 enveloping algebra -- Conjugation -- Application : Duflo's Theorem -- Relative Dirac Operators -- Harish-Chandra projections -- Enveloping algebras -- Clifford algebras -- Quantum Weil algebras -- Applications to reductive Lie algebras -- Notation -- Harish-Chandra projections -- Harish-Chandra projection for U(g) -- Harish-Chandra projection of the quadratic Casimir -- Harish-Chandra projection for Cl(g) -- Equal rank subalgebras -- The kernel of Dv -- q-dimensions -- The shifted Dirac Operator -- Dirac induction -- Central extensions of compact Lie groups -- Twisted representations -- The p-representation of g as a twisted representation of G -- Definition of the induction map -- The kernel of DM -- D(g, ...) as a geometric Dirac Operator -- Differential Operators on homogeneous Spaces -- Dirac Operators on manifolds -- Linear connections -- Principal connections -- Dirac Operators -- Dirac Operators on homogeneous Spaces -- The Hopf-Koszul-Samelson Theorem -- Lie algebra cohomology -- Lie algebra homology -- Definition and basic properties -- Schouten bracket -- Lie algebra homology for reductive Lie algebras -- Hopf algebra structure on (ĝ)g -- Primitive elements -- Hopf-Koszul-Samelson Theorem -- Consequences of the Hopf-Koszul-Samelson Theorem -- Transgression Theorem -- The Clifford algebra of a reductive Lie algebra -- Cl(g) and the p-representation -- Relation with extremal projectors -- Theisomorphism... -- The p-decomposition of elements... -- The space Hom... -- The Space Hom... -- The Harish-Chandra projection of ... -- Relation with the principal TDS -- Appendix A : Graded and filtered super spaces -- Super vector spaces -- Graded super vector spaces -- Filtered super vector spaces -- Appendix B : Reductive Lie algebras -- Definitions and basic properties -- Cartan subalgebras -- Representation theory of sl(2, C) -- Roots -- Simple roots -- TheWeylgroup -- Weyl Chambers -- Weights of representations -- Highest weight representations -- Extremal weights -- Multiplicity computations -- Appendix C : Background on Lie groups -- Preliminaries -- Group actions on manifolds -- The exponential map -- The vector field ... -- Maurer-Cartan forms -- Quadratic Lie groups -- References -- Index.
Subject Clifford algebras.
Lie algebras.
Lie groups.
ISBN 364236215X
9783642362156
OCLC number 825755201