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Weyl quantization and a symbol calculus for abelian groups

Published online by Cambridge University Press:  09 April 2009

N. J. Wildberger
N. J. Wildberger
Affiliation:
School of Mathematics UNSWSydney 2052Australia e-mail: n.wildberger@unsw.edu.au
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Abstract

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We develop a notion of a *-product on a general abelian group, establish a Weyl calculus for operators on the group and connect these with the representation theory of an associated Heisenberg group. This can all be viewed as a generalization of the familiar theory for R. A symplectic group is introduced and a connection with the classical Cayley transform is established. Our main application is to finite groups, where consideration of the symbol calculus for the cyclic groups provides an interesting alternative to the usual matrix form for linear transformations. This leads to a new basis for sl(n) and a decomposition of this Lie algebra into a sum of C*artan subalgebras.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 3 , June 2005 , pp. 323 - 338
Copyright
Copyright © Australian Mathematical Society 2005

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