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arXiv · 2406.02967

Atomic representations of R. Thompson's groups and Cuntz's algebra

Arnaud Brothier·Dilshan Wijesena
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Abstract

We continue to study Pythagorean unitary representation of Richard Thompson's groups $F,T,V$ and their extension to the Cuntz(-Dixmier) algebra. Any linear isometry from a Hilbert space to its direct sum square produces such. We focus on those arising from a finite-dimensional Hilbert space. We show that they decompose as a direct sum of a so-called diffuse part and an atomic part. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. In this article, we fully describe the atomic part: it is a finite direct sum of irreducible monomial representations arising from a precise family of parabolic subgroups.

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Arnaud Brothier, Dilshan Wijesena. 2024-06-05. Atomic representations of R. Thompson's groups and Cuntz's algebra. https://arxiv.org/abs/2406.02967

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