Syllabus/achievement requirements

Introduction to subfactors

The theory of subfactors was initiated by V. Jones in his seminal paper where he showed that inclusion of II1 factors admit intricate restriction on the relative size of algebras, or the index.

Soon afterwards fruitful connection with other branches of mathematics and mathematical physics emerged. On the one hand, the discovery of the Jones polynomial for knots led to the theory of quantum topological invariants. On the other, the curious restriction on the index motivated the development of the theory of dimension on C*-tensor categories. This tensor categorical point of view is also important in understanding the symmetry of conformal field theory.

In this course we begin with a review of basic theory of subfactors. On the algebraic and categorical side, our goal is to understand the standard invariants (also known as paragroups, λ-lattices, or planar algebras). On the analytic side, we aim to understand the concept of amenability, which is a fundamental concept in the theory of operator algebras and ergodic theory. Combining these two, and the Evans–Kishimoto type intertwining argument, we illustrate a new proof of Popa's classification due to Tomatsu.

In the remaining time we move on to more advanced topics, such as a recent operadic reconstruction of conformal nets by Tener.


  1. V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR MR696688 (84d:46097)
  2. Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111 (95f:46105)
  3. David E. Evans and Yasuyuki Kawahigashi, Quantum symmetries on operator algebras, Oxford Math- ematical Monographs, The Clarendon Press Oxford University Press, New York, 1998, Oxford Science Publications. MR 1642584 (99m:46148)

  4. Reiji Tomatsu, Centrally free actions of amenable C*-tensor categories on von Neumann algebras, preprint, arXiv:1812.04222.

  5. James E. Tener, Fusion and positivity in chiral conformal field theory, preprint, arXiv:1910.08257.

Published Dec. 10, 2019 11:07 AM - Last modified Jan. 7, 2020 2:52 PM