By Eckhard Meinrenken

This monograph offers an advent to the speculation of Clifford algebras, with an emphasis on its connections with the speculation of Lie teams and Lie algebras. The booklet starts off with a close presentation of the most effects on symmetric bilinear varieties and Clifford algebras. It develops the spin teams and the spin illustration, culminating in Cartan’s well-known triality automorphism for the crowd Spin(8). The dialogue of enveloping algebras encompasses a presentation of Petracci’s evidence of the Poincaré–Birkhoff–Witt theorem. this is often through discussions of Weil algebras, Chern--Weil idea, the quantum Weil algebra, and the cubic Dirac operator. The functions to Lie concept comprise Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The final a part of the booklet 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 touching on the Harish-Chandra projection for Clifford algebras to the significant sl(2) subalgebra. apart from those attractive functions, the ebook will function a handy and updated reference for historical past fabric from Clifford thought, correct for college students and researchers in arithmetic and physics.

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

**Example text**

Define an action of SU(2) × SU(2) on H by (X1 , X2 ) · X = X1 XX2−1 . This action preserves the bilinear form on H ∼ = R4 , and hence defines a homomorphism SU(2) × SU(2) → SO(4). The kernel of this homomorphism is the finite subgroup {±(I, I )} ∼ = Z2 . (Indeed, X1 XX2−1 = X for all X implies in particular −1 that X1 = XX2 X for all invertible X. ) Since dim SO(4) = 6 = 2 dim SU(2), and since SO(4) is connected, this homomorphism must be onto. Thus SO(4) = (SU(2) × SU(2))/{±(I, I )}. Similarly, identify R3 ∼ = {X ∈ H| tr(X) = 0} = span(I )⊥ .

Hence they descend to odd derivations ι(α) of Cl(V ; B) of filtration degree −1, given as k ι(α)(v1 · · · vk ) = (−1)i−1 α, vi v1 · · · vi · · · vk . 14) i=1 On the other hand, the Lie derivatives LA on T (V ) preserve the ideal I (V ; B) if and only if A ∈ o(V ; B), that is, B(Av1 , v2 ) + B(v1 , Av2 ) = 0 for all v1 , v2 . Under this condition, LA descends to an even derivation of filtration degree 0 k LA (v1 · · · vk ) = v1 · · · LA (vi ) · · · vk i=1 on the Clifford algebra. α = −A∗ α with A∗ the dual map.

Put differently, if A op is A with the opposite algebra structure a ·op b := ba, an anti-automorphism is an algebra isomorphism A → A op . The tensor algebra has a unique involutive anti-automorphism that is equal to the identity on V ⊆ T (V ). It is called the canonical anti-automorphism or transposition, and is given by (v1 ⊗ · · · ⊗ vk ) = vk ⊗ · · · ⊗ v1 . Since the transposition preserves the ideal I (V ) defining the exterior algebra ∧(V ), it descends to an anti-automorphism of the exterior algebra, φ → φ .

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

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