The classification of subfactors of index at most 5

Abstract

Abstract. A subfactor is an inclusion N ⊂ M of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action M G ⊂ M , 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's 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 '80s and early '90s. 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.

Chirality and principal graph obstructions

“…We refer the reader to [6,2,13] for the definition of the principal graphs (Γ + , Γ − ) of P • . If there is only one projection P in the equivalence class [P ] corresponding to a vertex of Γ ± , then we identify [P ] with P .…”

“…We rapidly recall the notions of supertransitivity and annular multiplicities for subfactor planar algebras and potential principal graphs following [11,12,20,13].…”

“…Haagerup classified principal graphs of subfactors with index in the range (4, 3 + √ 3) [7], and recently, subfactor planar algebras with index less than 5 were completely classified [19,20,8,22]. The recent survey [13] provides an excellent introduction to the subfactor classification program, along with state-of-the-art information on its progress.…”

Singly generated planar algebras of small dimension, Part III

“…Recently the classification has been extended up to index 5, and even beyond. See [JMS14,MS12,MPPS12,IJMS12,PT12]. Such classifications would not be possible without the reduction of the subfactor problem to an essentially combinatorial one.…”

An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem

“…[1,3,5,19]) constraints in hand, it has proved possible to enumerate all possible principal graphs for subfactors with small index. This approach was pioneered by Haagerup [7], and more recently continued, resulting in a classification of subfactors up to index 5 [8,11,15,17,21].…”