Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 687on the spinorss + =⎛⎝s + 1⎞⎠ and t + =⎛⎝t + 1⎞⎠ ∈ S + .s + 2t + 2The above Hermitian form is Spin(4)−invariant.The dual vector space S +∗ consists of complex linear functionalsφ : S + −→ C.S +∗ is generated by the dual complex basis{ 1 ∗ C(V ) , (w 1 ∧ w 2 ) ∗ }, which satisfies1 ∗ C(V ) ( 1 C(V ) ) = 1, 1∗ C(V ) ( w 1 ∧ w 2 ) = 0,(w 1 ∧ w 2 ) ∗ ( 1 C(V )) = 0, (w 1 ∧ w 2 ) ∗ ( w 1 ∧ w 2 ) = 1.There is a Hermitian Riesz representationwith the following identification⎛ ⎞⎝s + 1S + ∼=−→ S +∗ ,⎛⎠ ↦→ 〈 ⎝s + 1⎞⎠ , · 〉.s + 2s + 2The Hermitian Riesz representation S + −→ S +∗ is given by⎛ ⎞ ⎛ ⎞∗⎝s + 1⎠ ↦→ ⎝s + 1⎠ ,which implies⎛⎝s + 1s + 2s + 2⎞⎠∗⎛= 〈 ⎝s + 2s + 1s + 2⎞⎠ , · 〉.By using the Hermitian Riesz representation, we haveS + ⊗ S + ∼ = End( S + ).Also, by using the Hermitian Riesz representation, we haveS + ⊗ S + ∼ = Λ0C ⊕ Λ +C .