Ivancevic_Applied-Diff-Geom

690 Applied Differential Geometry: A Modern IntroductionNow consider the negative spinors S − . With respect to the standardbasis { w 1 , w 2 } we can define a similar Spin(4)−invariant symplecticform which takes the value{s − , t − } = s − 1 t− 2 − s− 2 t− 1on the spinorss − =⎛⎝s − 1⎞⎠ and t − =⎛⎝t − 1⎞⎠ ∈ S − .s − 2t − 2We have a similar symplectic Riesz representation on S − :with the following identification⎛ ⎞⎝s − 1S − ∼=−→ S −∗⎛⎠ ↦→ { ⎝s − 1⎞⎠ , ·},which, as in S + , satisfies⎛⎝s − 1s − 2s − 2⎞⎠∗⎛= { ⎝s − 2s − 2−s − 1⎞⎠ , ·}.As in S + , by using the symplectic Riesz representation, we have:S − ⊗ S − ∼ = End( S − ),S − ⊗ S − ∼ = Λ0C ⊕ Λ −C ,S + ⊗ S − ∼=−→ Hom( S + , S − ),S + ⊗ S − ∼ = Λ1C .Similarly, by interchanging S + and S − , and by using the symplecticRiesz representation, we haveS − ⊗ S + ∼ = Hom( S − , S + ),S − ⊗ S + ∼ = Λ1C .