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Quantum deformations of the Lorentz group. The Hopf-algebra level

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Quantum deformations of the Lorentz group. The Hopf-algebra level COMPOSITIO MATHEMATICA S. L.WORONOWICZ S. ZAKRZEWSKI Quantumdeformations of the Lorentz group. TheHopf-algebra level Compositio Mathematica, tome 90, no 2 (1994), p. 211-243. <http://www.numdam.org/item?id=CM_1994__90_2_211_0> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 211 Quantum deformations of the Lorentz group. The Hopf *-algebra level S. L. WORONOWICZ* Department of Mathematical Methods in Physics, University of Warsaw, Hoza 74, 00-682 Warsaw, Poland and S. ZAKRZEWSKI~~ Arnold Sommerfeld Institute, TU Clausthal Leibnizstr. 10, W-3392 Clausthal-Zellerfeld, Germany Received 17 September 1991; accepted in final form 10 December 1992 Abstract. Three properties characteristic for the Lorentz group are selected and all quantum groups with the same poperties are found. As a result, a number of one, two and three parameter quantum deformations of the Lorentz group is discovered. The deformations described in [1] and [2] are among them. Only the Hopf *-algebra level is discussed. Compositio Mathematica 90: 211-243, 1994. (Ç) 1994 Kluwer Academic Publishers. Printed in the Netherlands. 0. Introduction The existence of several different quantum deformations of the Lorentz group (cf. [1], [2]) raises the question of their classification. In this paper we give a complete answer to this question. More precisely we describe (on the Hopf *-algebra level) all quantum groups of 2 x 2

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