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

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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