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The µdeformed SegalBargmann transform as a Hall type transform
"... We present an explanation of how the µdeformed SegalBargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of SegalBargmann analysis. This explan ..."
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We present an explanation of how the µdeformed SegalBargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of SegalBargmann analysis. This explanation relates the µdeformed SegalBargmann transforms to the generalized SegalBargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the µdeformed SegalBargmann transform can be understood as Hall type transforms. In particular, we define a µdeformation of Hall’s “Version C ” generalized SegalBargmann transform which is then shown to be a µdeformed convolution with a µdeformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall. Keywords: SegalBargmann analysis, heat kernel analysis, µdeformed quantum mechanics.
Symmetry, Integrability and Geometry: Methods and Applications Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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 prog ..."
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Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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.
DUNKL OPERATORS AS COVARIANT DERIVATIVES IN A QUANTUM PRINCIPAL BUNDLE
"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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 prog ..."
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Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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. 1.
Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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 prog ..."
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Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional 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.