The classification of subfactors of index at most 5

Abstract

A subfactor is an inclusion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N subset-of upper M"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N \subset M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript upper G Baseline subset-of upper M"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>G</mml:mi> </mml:msup> <mml:mo> ⊂ </mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M^G \subset M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There is a measure of size of a subfactor, called the index. Remarkably, the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the 1980s and early 1990s. The possible index values above 4 are not quantized, but once you exclude a certain family, it turns out that again the possibilities are quantized. Recently, the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.