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13) To prove the statement about the subgroup G of P GL2 (A), it is sufficient to note that N is a normal subgroup of GL1 (A) × GL1 (A). We finish this discussion by recording one more simple fact. ab . ba If, for some involution f ∈ A, bf + a is invertible, then the image of f under T is an involution. 14. Let T ∈ G. Assume that T is represented by a matrix T = Proof. We use the formula f → (af +b)(bf +a)−1 , which is independent of the choice of representative of T in GL2 (A). Because f −1 = f , this can be written as f → (af +b)((b+ af −1 )f )−1 = (af + b)f (af + b)−1 , which immediately gives [(af + b)(bf + a)−1 ]2 = 1.

Using Eq. (95) we obtain [W ν (a), ν ε (x)] N = L/2 lim j=1 ε ↓0 1 × 1 + (ν 2 − 1)a∂y + O(a2 ) (·)j 2 × × −L/2 dyδxj ,ε˜j (y) (eiν[fy+a,ε ˜ −f˜y,ε ] × )× ∗ ν ε (x). By a simple computation we see that this equals N ˜ νε (x; aej ) + iπν(ν 2 − 1)a[ρ˜εj (yj + a) − ρ˜εj (yj )] ∗ ˜ νε (x; aej ) j=1 N ν 2 (ν 2 − 1) + j,k=1 π π π ν cot (ykj + a) − cot (ykj ) ˜ ε (x; aej ) + O(a3 ), L L L j=k where yjk = yj − yk + isgn(k − j)(εj + εk ) and ˜ νε (x; aej ) := φνε (x1 ) · · · φ˜ νε (xj ; a) · · · φνε (xN ) 1 j N with φ˜ νε (x; a) defined in Eq.

Phys. : Exact results for a quantum many body problem in one-dimension. Phys. Rev. A4 2019 (1971); ibid. , J. Math. Phys. 10 2197 (1969) [PS] [RS1] [RS2] [R] [S] [Se] [SeW] [Sk] [St] [StW] [Su] Communicated by D. C. Brydges Commun. Math. Phys. 201, 35 – 60 (1999) Communications in Mathematical Physics © Springer-Verlag 1999 M¨obius Transformations in Noncommutative Conformal Geometry P. J. M. Bongaarts1, , J. O. Box 9506, 2300 RA Leiden, The Netherlands. nl 2 Department of Mathematics, University of Exeter, North Park Road, Exeter EX4 5HE, UK.

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