Toroidal orbifolds of Z3Z3 and Z6Z6 symmetries of noncommutative tori

The Hexic transform ρ   of the noncommutative 2-torus AθAθ is the canonical order 6 automorphism defined by ρ(U)=Vρ(U)=V, ρ(V)=e−πiθU−1Vρ(V)=e−πiθU−1V, where U, V   are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUVVU=e2πiθUV. The Cubic transform is κ=ρ2κ=ρ2. These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C⁎-algebras Aθρ, Aθκ are noncommutative Z6Z6, Z3Z3 toroidal orbifolds, respectively. For a large class of irrationals θ   and rational approximations p/qp/q of θ, a projection e   of trace q2θ−pqq2θ−pq is constructed in AθAθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθeeAθe contains a Hexic invariant q×qq×q matrix algebra MM whose unit is e and such that the cut downs eUe, eVe   are approximately inside MM. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t)E(t) of C∞C∞-projections of the continuous field {At}0