slides

Example: Twisting by a Torus Action
Take H = A(T 2 ), generated by t1, t2 with tjt ∗ j = t∗ j tj = 1,
Define a cocycle by
∆(tj) = tj ⊗ tj, S(tj) = t ∗ j , ɛ(tj) = 1.
F (ti, ti) = 1, F (t1, t2) := exp( 1
2 iπθ),
extended as a Hopf bicharacter. Then H = HF as a Hopf algebra, but the
category of H-comodules is twisted.
Example: The coproduct ∆ : H → H ⊗ H makes H into an H-comodule algebra
in the category H C. The comodule-twisted torus has algebra relations
t1 ·F t2 = F (t1, t2)t1t2 = F 2 (t1, t2)t2 ·F t1 = µt2 ·F t1, µ = exp(iπθ)
i.e. we get the noncommutative torus A(T 2 θ ) as an algebra in HF C.
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 13 / 25