Abstract
The concept of quantum groups is important for the study of the quantum Yang-Baxter equations, Drinfeld [2], Jimbo [4], Manin [5] and others. On the other hand, Woronowicz [10] introduced the concept of compact matrix pseudogroups through the study of the dual object of groups. As pointed out by Rosso in [8], these two concepts are related to each other as quantum Lie algebras and quantum Lie groups. In this talk we want to indicate that the ideas of Kac algebras studied by Takesaki [9] and Enock and Schwartz [3] et al. are helpful for the study of quantum groups. As a result we can give a geometric interpretation for a q-analogue of a certain class of special functions, which has been a long standing problem of q- analogues.
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References
Andrews, G.E. and R. Askey, Enumeration of partitions. The role of Eulerian series and the q-orthogonal polynomials, Higher Combinatorics, edited by M. Aigner, 2–26, Reidel, Dordrecht, Holland, 1977.
Drinfel’d, V.G., Quantum groups, Proc. of International Congress of Math., Berkeley, California, USA, 1986, 798–820.
Enock, M. and J.M. Schwartz, Une dualité dans algè bres de von Neumann, Bull. Soc. Math. France Suppl. Mémoire, 44 (1975), 1–144.
Jimbo, M., A q-difference analogue of U(g) and Yang-Baxter equation, Lett. in Math. Phys. 10 (1985), 63–69.
Manin, Y.I., Some remarks on Koszul algebras and quantum groups, Ann. de l’Inst. Fourier, 1987 (Colloque en l’honeur de J.-L. Koszul).
Masuda, T., K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno, Representations of quantum groups and a q-analogue of polynomials, C.R. Acad. Sc. Paris, 307 (1988), 559–564.
Poldes, P., Quantum spheres, Lett, in Math. Phys. 14 (1987), 193–202.
Rosso, M., Comparison des groupes SU(2)quantiques de Drinfeld et de Woronowicz, C.R. Acad. Sc. Paris 304 (1987), 323–326.
Takesaki, M., Duality and von Neumann algebras, Lecture Notes in Math. 247 (1972), 665–785, Springer-Verlag, Berlin.
Woronowicz, S.L., Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. RIMS, Kyoto Univ. 23 (1987), 117–181.
Woronowicz, S.L., Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613–665.
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Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K. (1991). Representation of Quantum Groups. In: Araki, H., Kadison, R.V. (eds) Mappings of Operator Algebras. Progress in Mathematics, vol 84. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0453-4_4
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