Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 040, 29 pages      arXiv:1108.3769      http://dx.doi.org/10.3842/SIGMA.2013.040

Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

Micho Đurđevich a and Stephen Bruce Sontz b
a) Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, CP 04510, Mexico City, Mexico
b) Centro de Investigación en Matemáticas, A.C. (CIMAT), Jalisco s/n, Mineral de Valenciana, CP 36240, Guanajuato, Gto., Mexico

Received November 01, 2012, in final form May 17, 2013; Published online May 30, 2013

Abstract
A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.

Key words: Dunkl operators; quantum principal bundle; quantum connection; quantum curvature; Coxeter groups.

pdf (455 kb)   tex (38 kb)

References

  1. Ben Saïd S., Kobayashi T., Ørsted B., Laguerre semigroup and Dunkl operators, Compos. Math. 148 (2012), 1265-1336, arXiv:0907.3749.
  2. Cherednik I., Generalized braid groups and local r-matrix systems, Soviet Math. Dokl. 40 (1990), 43-48.
  3. Cherednik I., A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411-431.
  4. Chouchene F., Gallardo L., Mili M., Les équations de la chaleur et de Poisson pour le laplacien généralisé de Jacobi-Dunkl, C. R. Math. Acad. Sci. Paris 341 (2005), 179-184.
  5. Connes A., Non-commutative differential geometry, Publ. Math. Inst. Hautes Études Sci. 62 (1985), 41-144.
  6. De Bie H., Ørsted B., Somberg P., Souček V., Dunkl operators and a family of realizations of osp(1|2), Trans. Amer. Math. Soc. 364 (2012), 3875-3902, arXiv:0911.4725.
  7. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  8. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  9. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  10. Dunkl C.F., de Jeu M.F.E., Opdam E.M., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256.
  11. Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70-108.
  12. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  13. Đurđevich M., Characteristic classes of quantum principal bundles, Algebras Groups Geom. 26 (2009), 241-341, q-alg/9507017.
  14. Đurđevich M., Geometry of quantum principal bundles. I, Comm. Math. Phys. 175 (1996), 457-520, q-alg/9507019.
  15. Đurđevich M., Geometry of quantum principal bundles. II. Extended version, Rev. Math. Phys. 9 (1997), 531-607, q-alg/9412005.
  16. Đurđevich M., Geometry of quantum principal bundles. III, Algebras Groups Geom. 27 (2010), 247-336.
  17. Etingof P., Calogero-Moser systems and representation theory, Zürich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2007.
  18. Grove L.C., Benson C.T., Finite reflection groups, Graduate Texts in Mathematics, Vol. 99, 2nd ed., Springer-Verlag, New York, 1985.
  19. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  20. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Interscience Publishers, New York - London, 1963.
  21. Opdam E.M., Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, Vol. 8, Mathematical Society of Japan, Tokyo, 2000.
  22. Oziewicz Z., Relativity groupoid instead of relativity group, Int. J. Geom. Methods Mod. Phys. 4 (2007), 739-749.
  23. Prugovecki E., Quantum geometry. A framework for quantum general relativity, Fundamental Theories of Physics, Vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1992.
  24. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
  25. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  26. Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
  27. Serre J.P., Lie algebras and Lie groups, W.A. Benjamin, Inc., New York - Amsterdam, 1965.
  28. Sontz S.B., On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel, Math. Z. 269 (2011), 9-28.
  29. Sutherland B., Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev. A 5 (1972), 1372-1376.
  30. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  31. Woronowicz S.L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125-170.
  32. Woronowicz S.L., Pseudospaces, pseudogroups and Pontriagin duality, in Mathematical Problems in Theoretical Physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979), Lecture Notes in Phys., Vol. 116, Springer, Berlin, 1980, 407-412.

Previous article  Next article   Contents of Volume 9 (2013)