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Clifford algebras and lie theory /

Bibliographic Details
Main Author: Meinrenken, Eckhard (Author)
Format: Book
Language:English
Published: Heidelberg : Springer, [2013]
Series:Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 58.
Subjects:
Clifford algebras.
Lie algebras.
Lie groups.
Table of Contents:
  • Convention
  • List of Symbols
  • Symmetric 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 contractio̧ns
  • 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 ^g*
  • g-homotopies
  • The Weil algebra
  • Chern-Weil homomorphisms
  • The non-commutative Weil algebra W̃g
  • 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 (^g)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.