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Spectral Measures of Small Index Principal Graphs

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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 ≤  4 the spectral measure of U can be averaged by using the map u→ u −1, and we get a probability measure \(\varepsilon\) on the unit circle which does not depend on U. We find explicit formulae for this measure \(\varepsilon\) for the principal graphs of subfactors with index ≤ 4, the (extended) Coxeter-Dynkin graphs of type A, D and E. The moment generating function of \(\varepsilon\) is closely related to Jones’ Θ-series.

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Authors and Affiliations

  1. Department of Mathematics, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Teodor Banica

  2. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Dietmar Bisch

Authors
  1. Teodor Banica
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  2. Dietmar Bisch
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Correspondence to Dietmar Bisch.

Additional information

Communicated by Y. Kawahigashi

D.B. was supported by NSF under Grant No. DMS-0301173.

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Banica, T., Bisch, D. Spectral Measures of Small Index Principal Graphs. Commun. Math. Phys. 269, 259–281 (2007). https://doi.org/10.1007/s00220-006-0122-1

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  • DOI: https://doi.org/10.1007/s00220-006-0122-1

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