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Spectral measures of small index principal graphs

Teodor Banica·Dietmar Bisch
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Abstract

The principal graph $X$ of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If $Δ$ is the adjacency matrix of $X$ we consider the equation $Δ=U+U^{-1}$. When $X$ has square norm $\leq 4$ the spectral measure of $U$ can be averaged by using the map $u\to u^{-1}$, and we get a probability measure $ε$ on the unit circle which does not depend on $U$. We find explicit formulae for this measure $ε$ for the principal graphs of subfactors with index $\le 4$, the (extended) Coxeter-Dynkin graphs of type $A$, $D$ and $E$. The moment generating function of $ε$ is closely related to Jones' $Θ$-series.

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Teodor Banica, Dietmar Bisch. 2006-05-26. Spectral measures of small index principal graphs. https://doi.org/10.1007/s00220-006-0122-1

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