# Communications In Mathematical Physics - Volume 281 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Connes, M. Dubois-Violette Thus the characteristic variety consists in the six lines parameters x j by j given in terms of free {(0, x1 , x2 , 0)}, {(x0 , 0, 0, x3 )}, {(x0 , x1 , i x1 , i x0 )}, {(x0 , x1 , −i x1 , i x0 )}, {(x0 , x1 , i x1 , −i x0 )}, {(x0 , x1 , −i x1 , −i x0 )}. 19) The correspondence σ is the identity on the first two lines. It preserves globally the other j and induces on each of them the rational automorphism given by multiplication by e±2iϕ . 10. Edge α ⊥ β and (n 1 , n 2 ) = (odd, odd).

Q = 1 in Uq (su(2))hom ). Case 7 is the θ -deformation studied in Part I [15] and in [19] while Case 8 (anti θ -deformation) is obtained by α1 -duality (cf. Sect. 7) from Case 7. Case 9 is very singular: 3 relations are missing. Case 10 is obtained by α3 -duality (cf. Sect. 7) from Case 11 which is the ordinary algebra of polynomials with 4 indeterminates C[x 0 , x 1 , x 2 , x 3 ](classical case). Note that the reality conditions above correspond to the hermiticity of the x µ and not of the Sklyanin generators Sµ .

2. The critical set C of X is given by C = W (C+ ) ∪ W (C− ) ∪ W (P), P= π π π , , . 2 2 2 One checks this directly in the ψ variables. 7) and we let D = {ϕ, δ(ϕ) = 0}. 8) By construction D is invariant under the group S3 of permutations of the ϕ j ’s. 3. Let ϕ ∈ / C, then there exists g ∈ W such that gϕ ∈ / D. Proof. Let us first show that the conclusion holds if one of the ϕ j vanishes (we always work modulo π and all equalities below have this meaning). e. that ϕ = (ϕ1 , ϕ2 , 0). Then if ϕ1 = ϕ2 one is in C thus we can assume that ϕ1 = ϕ2 .