Introduction to Subfactors

Abstract

Subfactors have been a subject of considerable research activity for about fifteen years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. A new pictorial approach to subfactors is presented in a late chapter.

The quantum algebra of partial Hadamard matrices

“…In the square case, M = N, the idea is that any complex Hadamard matrix H ∈ M N (T) produces a certain quantum subgroup G ⊂ S + N of Wang's quantum permutation group [17]. This construction, inspired from [9], [13], and heavily relying on Woronowicz's work in [19], was axiomatized in [1], [3]. One problem here, still open, is that of finding precise relations between the invariants of G and the invariants of H.…”

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“…However, the construction still keeps secrets. Ocneanu compactness ([12], [9]) provides a method of computing the standard invariant of R C ⊂ R and, in particular, the principal graph ([8], [4]). (The principal graph Γ is a possibly infinite graph with a distinguished root vertex * and an eigenvector τ with eigenvalue the Jones index [R : R C ].…”

“…there is a well-known way to construct a subfactor R C ⊂ R of the hyperfinite II 1 factor from C (for commuting squares see [14], [4], [9]; for the construction see [19], [9]). This construction is quite general.…”

“…Let k and p be positive integers and U a unitary matrix in M p (C) ⊗ M k (C). A vertex model commuting square ( [9]) is a commuting …”

“…From this point it is straightforward to describe Γ in terms of the representation theory of subgroups of G. As an example, we apply our theory to the case k = 2 and show that the principal graphs of the Krishnan-Sunder subfactors of index 4 are A (1) 2n−1 , n ≥ 1, and D (1) n , n ≥ 4. Our theory also yields a short proof of Theorem 37 in [10], which describes the principal graph of a particular infinite depth Krishnan-Sunder subfactor.We remark that if H and G are the group and graph of the index p 2 KrishnanSunder subfactor arising from (1.1), then G and H are bipartite generalizations of the Cayley graphs of G and H, respectively (see Example 6.2.2 of [9] for an example of this type of Cayley graph).In [10], Krishnan and Sunder state that they believe the principal graphs of their two infinite depth subfactors are trees. We show that this is not the case, and in particular that the principal graph of the irreducible infinite depth subfactor is not a tree.…”