New PDF release: Clifford Algebras and Lie Theory (December 2012 version)

By Eckhard Meinrenken

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Additional resources for Clifford Algebras and Lie Theory (December 2012 version)

Example text

The kernel of this homomorphism is the finite subgroup {±(I, I)} ∼ = Z2 . (Indeed, X1 XX2−1 = X for all X implies in particular that −1 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)⊥ . The conjugation action of SU(2) on H preserves this subspace; hence we obtain a group homomorphism SU(2) → SO(3). The kernel of this homomorphism is Z2 ∼ = {±I} ⊆ SU(2).

Let {·, ·} be the induced Lie bracket on ∧2 (V ) so that [q(λ ), q(λ )] = q({λ , λ }). The transformation v → Aλ (v) = [q(λ ), v] defines an element Aλ ∈ o(V ; B), and the map ∧2 (V ) → o(V ; B), λ → Aλ is a Lie algebra homomorphism. One has LAλ = [q(λ ), ·] as derivations of Cl(V ; B). 40 2 Clifford algebras Proof. By definition, Aλ (v) = [q(λ ), v] = −2ι(B (v))q(λ ). Hence Aλ (v) = −2ι(B (v))λ since the quantization map intertwines the contractions of the exterior and Clifford algebras. We have Aλ ∈ o(V ; B) since B(Aλ (v), w) = −2ι(B (w))Aλ (v) = −2ι(B (w))ι(B (v))λ is anti-symmetric in v, w.

Suppose V1 ,V2 are two vector spaces. Then ∧(V1 ) ⊗ ∧(V2 ) (tensor product of graded super algebras) with the natural inclusion of V1 ⊕ V2 satisfies the universal property of the exterior algebra over V1 ⊕V2 . Hence the morphism of graded super algebras ∧(V1 ⊕V2 ) → ∧(V1 ) ⊗ ∧(V2 ) intertwining the two inclusions is an isomorphism. As a special case, ∧(Kn ) = ∧(K) ⊗ · · · ⊗ ∧(K). 3 Derivations The space Der(∧(V )) of derivations of the graded super algebra ∧(V ) is a left module over ∧(V ), since ∧(V ) is commutative.

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Clifford Algebras and Lie Theory (December 2012 version) by Eckhard Meinrenken

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