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SIGMA 9 (2013), 040, 29 pages arXiv:1108.3769
https://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.
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References
- Ben Saïd S., Kobayashi T., Ørsted B., Laguerre semigroup and Dunkl
operators, Compos. Math. 148 (2012), 1265-1336,
arXiv:0907.3749.
- Cherednik I., Generalized braid groups and local r-matrix systems,
Soviet Math. Dokl. 40 (1990), 43-48.
- Cherednik I., A unification of Knizhnik-Zamolodchikov and Dunkl
operators via affine Hecke algebras, Invent. Math. 106
(1991), 411-431.
- 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.
- Connes A., Non-commutative differential geometry, Publ. Math. Inst.
Hautes Études Sci. 62 (1985), 41-144.
- 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.
- de Jeu M.F.E., The Dunkl transform, Invent. Math. 113
(1993), 147-162.
- 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.
- Dunkl C.F., Differential-difference operators associated to reflection groups,
Trans. Amer. Math. Soc. 311 (1989), 167-183.
- 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.
- Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups,
Proc. London Math. Soc. (3) 86 (2003), 70-108.
- Dunkl C.F., Xu Y., Orthogonal polynomials of several variables,
Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge
University Press, Cambridge, 2001.
- Đurđevich M., Characteristic classes of quantum principal bundles,
Algebras Groups Geom. 26 (2009), 241-341,
q-alg/9507017.
- Đurđevich M., Geometry of quantum principal bundles. I,
Comm. Math. Phys. 175 (1996), 457-520,
q-alg/9507019.
- Đurđevich M., Geometry of quantum principal bundles. II. Extended
version, Rev. Math. Phys. 9 (1997), 531-607,
q-alg/9412005.
- Đurđevich M., Geometry of quantum principal bundles. III,
Algebras Groups Geom. 27 (2010), 247-336.
- Etingof P., Calogero-Moser systems and representation theory, Zürich
Lectures in Advanced Mathematics, European Mathematical Society (EMS),
Zürich, 2007.
- Grove L.C., Benson C.T., Finite reflection groups, Graduate Texts in
Mathematics, Vol. 99, 2nd ed., Springer-Verlag, New York, 1985.
- Humphreys J.E., Reflection groups and Coxeter groups, Cambridge
Studies in Advanced Mathematics, Vol. 29, Cambridge University Press,
Cambridge, 1990.
- Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I,
Interscience Publishers, New York - London, 1963.
- Opdam E.M., Lecture notes on Dunkl operators for real and complex reflection
groups, MSJ Memoirs, Vol. 8, Mathematical Society of Japan, Tokyo,
2000.
- Oziewicz Z., Relativity groupoid instead of relativity group, Int. J.
Geom. Methods Mod. Phys. 4 (2007), 739-749.
- Prugovecki E., Quantum geometry. A framework for quantum general
relativity, Fundamental Theories of Physics, Vol. 48, Kluwer
Academic Publishers Group, Dordrecht, 1992.
- 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.
- Rösler M., Generalized Hermite polynomials and the heat equation for
Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542,
q-alg/9703006.
- Rösler M., Voit M., Markov processes related with Dunkl operators,
Adv. in Appl. Math. 21 (1998), 575-643.
- Serre J.P., Lie algebras and Lie groups, W.A. Benjamin, Inc., New York -
Amsterdam, 1965.
- Sontz S.B., On Segal-Bargmann analysis for finite Coxeter groups and its
heat kernel, Math. Z. 269 (2011), 9-28.
- Sutherland B., Exact results for a quantum many-body problem in
one-dimension. II, Phys. Rev. A 5 (1972), 1372-1376.
- Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys.
111 (1987), 613-665.
- Woronowicz S.L., Differential calculus on compact matrix pseudogroups (quantum
groups), Comm. Math. Phys. 122 (1989), 125-170.
- 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.
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