slides

Quaternionic Structure
Note that the A(C 4 )-valued matrix
Ψ =
tr z1 z2 z3 z4
−z ∗ 2 z ∗ 1 −z ∗ 4 z ∗ 3
is made from a pair of quaternion-valued functions.
This quaternion structure is encoded on A(C4 ) by the ∗-algebra map
J : A(C 4 ) → A(C 4 ), J z1 z2 z3
∗
z4 = −z2 z∗ 1 −z∗ 4 z∗
3 .
This gives an identification of C 4 with H 2 .
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 7 / 25