7 research outputs found
Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
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
Generalized Noiseless Quantum Codes utilizing Quantum Enveloping Algebras
A generalization of the results of Rasetti and Zanardi concerning avoiding
errors in quantum computers by using states preserved by evolution is
presented. The concept of dynamical symmetry is generalized from the level of
classical Lie algebras and groups to the level of dynamical symmetry based on
quantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural
connection is proved between states preserved by representations of a quantum
group and states preserved by evolution with dynamical symmetry of the
appropriate universal enveloping algebra. Illustrative examples are discussed.Comment: 10 pages, LaTeX, 2 figures Postscrip