Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

Details Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

2011-08-18

arxiv.org/abs/1108.3769v1

Physics

Mathematical Physics

30 pages

Scientific paper

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 commutivity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.

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