Clifford Algebras and Lie Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

By Eckhard Meinrenken

This monograph offers an advent to the idea of Clifford algebras, with an emphasis on its connections with the idea of Lie teams and Lie algebras. The booklet starts off with an in depth presentation of the most effects on symmetric bilinear kinds and Clifford algebras. It develops the spin teams and the spin illustration, culminating in Cartan’s well-known triality automorphism for the gang Spin(8). The dialogue of enveloping algebras contains a presentation of Petracci’s facts of the Poincaré–Birkhoff–Witt theorem.

This is by means of discussions of Weil algebras, Chern--Weil concept, the quantum Weil algebra, and the cubic Dirac operator. The purposes to Lie idea contain Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The final a part of the e-book is an account of Kostant’s constitution concept of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his interesting conjecture bearing on the Harish-Chandra projection for Clifford algebras to the vital sl(2) subalgebra.

Aside from those appealing purposes, the ebook will function a handy and updated reference for history fabric from Clifford conception, appropriate for college kids and researchers in arithmetic and physics.

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Evidence The facts is comparable to that of Proposition 6. 6. the most aspect is that the 2 morphisms c0 , c1 : W g → W g ⊗ W g given by means of c0 (w) = w ⊗ 1, c1 (w) = 1 ⊗ φ(w) are the attribute homomorphisms for the 2 average connections on W g ⊗ W g. for that reason they're g-homotopic (Property three above), and their compositions with W g ⊗ W g → W g, w ⊗ w → wsym(w ) are back g-homotopic. Corollary 6. 2 The quotient map W g → W g induces an isomorphism Hbas (W g) = (Sg∗ )g . 156 6 Weil algebras 6. 12 Equivariant cohomology of g-differential areas Definition 6.

Submodules, quotient modules. A submodule of a Cl(V )-module E is an excellent subspace E1 that's reliable less than the module motion. consequently the quotient E/E1 turns into a Cl(V )-module in an noticeable approach. A Cl(V )-module E is named irreducible if there are not any submodules except E and {0}. 2. Direct sum. The direct sum of 2 Cl(V )-modules E1 , E2 is back a Cl(V )module, with ρE1 ⊕E2 = ρE1 ⊕ ρE2 . three. twin modules. If E is any Clifford module, then the twin house E ∗ = Hom(E, okay) turns into a Clifford module, with module constitution outlined when it comes to the canonical anti-automorphism of Cl(V ) through ρE ∗ (x) = ρE (x )∗ , x ∈ Cl(V ).

275 275 277 279 Appendix B Reductive Lie algebras . . B. 1 Definitions and simple houses . B. 2 Cartan subalgebras . . . . . . . . B. three illustration idea of sl(2, C) B. four Roots . . . . . . . . . . . . . . . B. five basic roots . . . . . . . . . . . B. 6 The Weyl team . . . . . . . . . B. 7 Weyl chambers . . . . . . . . . . B. eight Weights of representations . . . . B. nine maximum weight representations . B. 10 Extremal weights . . . . . . . . B. eleven Multiplicity computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 23 24 25 26 26 27 27 29 30 31 32 34 35 36 37 xiii xiv Contents 2. 2. 10 The Lie algebra q(∧2 (V )) . . . . . 2. 2. eleven A formulation for the Clifford product 2. three The Clifford algebra as a quantization . . . 2. three. 1 Differential operators . . . . . . . 2. three. 2 Graded Poisson algebras . . . . . . 2. three. three Graded tremendous Poisson algebras . . 2. three. four Poisson buildings on ∧(V ) . . . . . . . . . . . . . . . . .

Comment three. 10 (Restrictions) Any Cl(n)-module may be considered as a Cl(n − 1)module by way of limit. by way of measurement count number, one verifies: 1. If n is even, then the ungraded module Sn restricts to an immediate sum of the 2 non-isomorphic ungraded Cl(n − 1)-modules (given by means of the even and extraordinary half op of Sn ). the 2 Z2 -graded modules Sn and Sn either turn into isomorphic to the original Z2 -graded module over Cl(n − 1). 2. If n is peculiar, then the constraints of the 2 irreducible ungraded Cl(n)-modules to Cl(n − 1) are either isomorphic to Sn−1 , whereas the restrict of the irreducible op Z2 -graded Cl(n)-module is isomorphic to an instantaneous sum Sn−1 ⊕ Sn−1 .

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