The Haar state on the Vaksman-Soibelman quantum spheres
DOI:
https://doi.org/10.7146/math.scand.a-136693Abstract
In this note we present explicit formulae for the Haar state on the Vaksman-Soibelman quantum spheres. Our formulae correct various statements appearing in the literature and our proof is straightforward relying simply on properties of the modular automorphism group for the Haar state.
References
Blanchard, E. Déformations de $C^*$-algèbres de Hopf, Bull. Soc. Math. France 124 (1996), no. 1, 141–215. http://www.numdam.org/item?id=BSMF_1996__124_1_141_0
Hong, J. H., and Szymański, W., Quantum spheres and projective spaces as graph algebras, Comm. Math. Phys. 232 (2002), no. 1, 157–188. https://doi.org/10.1007/s00220-002-0732-1
Klimyk, A., and Schmüdgen, K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin 1997. https://doi.org/10.1007/978-3-642-60896-4
Nagy, G., A deformation quantization procedure for $C^*$-algebras, J. Operator Theory 44 (2000), no. 2, 369–411.
Nagy, G., On the Haar measure of the quantum $SU(N)$ group, Comm. Math. Phys. 153 (1993), no. 2, 217–228. http://projecteuclid.org/euclid.cmp/1104252678
Neshveyev, S., and Tuset, L., Compact quantum groups and their representation categories, Cours Spécialisés 20, Société Mathématique de France, Paris 2013.
Sheu, A. J. L., Compact quantum groups and groupoid $C^*$-algebras, J. Funct. Anal. 144 (1997), no. 2, 371–393. https://doi.org/10.1006/jfan.1996.2999
Soibelman, Y. S., and Vaksman, L. L., On some problems in the theory of quantum groups, Representation theory and dynamical systems, 3–55, Adv. Soviet Math., 9, Amer. Math. Soc., Providence, RI, 1992
Vaksman, L. L., and Soibelman, Y. S., Algebra of functions on the quantum group SU(n+1) and odd-dimensional quantum spheres, Algebra i Analiz 2 (1990), no. 5, 101–120; translation in Leningrad Math. J. 2 (1991), no. 5, 1023–1042.
Van Daele, A., The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3125–3128. https://doi.org/10.2307/2160670
Woronowicz, S. L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. http://projecteuclid.org/euclid.cmp/1104159726
Woronowicz, S. L., Compact quantum groups, Symétries quantiques (Les Houches, 1995), 845–884, North-Holland, Amsterdam 1998.
