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We have the following relation: 00100 0 0 0 1 0 G G = 1 0 0 0 0 , ∼ e e 0 1 0 0 0 00001 here ∼ at the lower left corner in the left hand side square means the changing of labels and replacing columns according to the labels. Now we get the gauge as usual. Note that gauge matrices for upside down edges are the same as those of normal position: ˜ d u G u e5 4,r G G 2,l d = u G 2,l 1 ∼ e e 1 2 3 (β √−2) 2β 4 2(β 4 +4) √ √ − 2(β 2 −2) β 2 (β 2 −1)(β 4 +4) = 2 β 2 (β √−2) (β 2√ −1) 2(β 4 +4) β 2 (β 2 −2) √ β 4 +4 √ 2 √ 2β (β 2 +1)(β 4 +4) u e 5 4,r 1 −(β −2) √ 4 √ 2(β +4) 2(β 2 −1) √ β 2 +1 2β 2 −1 2 0 √ 2 4 √β (β +4) 2(β 2 +1) √ 0 0 β 4 +4 2(β 2 +4) 4 √β +4 − β 2 (β 2 −2) = √ β 4 +4 √ −2 2 2 4 β (β −1)(β +4) 0 β 2 −2 2(β 2 −1) β 2 −2 2β 2 −1 β 4 +4 √ 2 − 2(β √−2) 4 (β 2 −1) √ β +4 −2 β 2 −2 2(β 2 −2) β 4 +4 √ √−1 2β 2 2 √β −2 β 4 −1 √ 2 β 2 √1 2(β 2 +4) β 4 +4 β2 β 4 +4 √ 2β 4 (2β 2 −1)(β √ +4) (β 2 −1) β 2 −1 √ β 2 (β 4 +4) 0 β 2 −1 −(β 2 −2) 2(β 2 −1) √1 2 √ β −1 2 √2β −1 β 4 −1 0 √−1 u e 5 4,r 1 , β 2 +1 √ 2 2 √β (β −2) − β 4 +4 √ 2β (2β 2 −1)(β 4 +4) 0 √ 2 β 0 √ −2 4 β 2 (β 2 −1)(β √ +4) (β 2 −1) β 2 −1 √ β 2 (β 4 +4) √ 2 β −1 β 2 −2 0 0 0 .

Orbifold subfactors from Hecke algebras. Commun. in Math. Phys. , de la Harpe, P. : Coxeter graphs and towers of algebras. : Principal graphs of subfactors in the index range 4 < [M : N ] < 3 + 2. Subfactors. Singapore: World Scientific, 1994, pp. 1–38 [HW] de la Harpe, P. : Operations sur les rayons spectraux de matrices symetriques entieres positives. C. R. Acad. Sci. : Numerical evidence for flatness of Haagerup’s connections. : Application of fusion rules to classification of subfactors. Publ.

The most celebrated one is the noncommutative torus, which played a significant role in the development of the subject [5]. The C*-algebra Tθ 2 is generated by two unitaries U, V subject to the commutation relation U V = e2πiθ V U . In particular, for θ = 0, this is just the C*-algebra of continuous functions on the two torus T2 and the noncommutative version can in a natural way be regarded as a deformation of this commutative C*-algebra. Actually this is the way that the noncommutative tori appear in physics (Hall effect), where the phase factor e2πiθ comes from the shift in the phase of an electron on a lattice in a transversal magnetic field.