Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
Mathematical Physics
2013-05-31 v2 Classical Analysis and ODEs
math.MP
Rings and Algebras
Abstract
A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space , 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.
Cite
@article{arxiv.1108.3769,
title = {Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle},
author = {Micho Durdevich and Stephen Bruce Sontz},
journal= {arXiv preprint arXiv:1108.3769},
year = {2013}
}