J. Aust. Math. Soc.  78 (2005), 323-338
Weyl quantization and a symbol calculus for abelian groups

N. J. Wildberger
  School of Mathematics
  Sydney 2052

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 $\mathbb{R}$. 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 $\operatorname{sl}(n)$ and a decomposition of this Lie algebra into a sum of Cartan subalgebras.
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