slides

Differential and Hodge Structure on S 4 θ
The coaction A(S 4 ) → H ⊗ A(S 4 ) is by isometries, i.e. it is an intertwiner
for the exterior derivative d : A(S 4 ) → Ω 1 (S 4 ).
This means that the differential calculus Ω • (S 4 ) is also an object in H C. So
we can deform it to get a differential calculus Ω • (S 4 θ ) on S4 θ .
Similarly, the Hodge operator ∗ : Ω 2 (S 4 ) → Ω 2 (S 4 ) is H-equivariant and so
it is a morphism in H C. Similarly for the map J : A(C 4 ) → A(C 4 ). Their
images under LF are a Hodge operator and a quaternion structure
∗θ : Ω 2 (S 4 θ ) → Ω 2 (S 4 θ ), J : A(C 4 θ) → A(C 4 θ).
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 15 / 25