Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

Micho Đurđevich ; Stephen Bruce Sontz

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

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.

Détail

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

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