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Discrete series of fusion algebras

Part of: Selfadjoint operator algebras Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

R. Srinivasan ,
V. S. Sunder  and
N. J. Wildberger
R. Srinivasan
Affiliation:
Indian Statistical Institute, R. V. College Post, Bangalore 560059, India e-mail: srinivasan.raman@labomath.univ-orleans.fr
V. S. Sunder
Affiliation:
The Institute of Mathematical Sciences, Taramani Chennai 600113, India e-mail: sunder@imsc.ernet.in
N. J. Wildberger
Affiliation:
University of New South Wales, Sydney NSW 2052, Australia e-mail: n.wildberger@unsw.edu.au
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Abstract

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We show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.

We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.

We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

MSC classification

Secondary: 46L37: Subfactors and their classification 43A65: Representations of groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 419 - 426
Copyright
Copyright © Australian Mathematical Society 2002

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