If $N \subset P,Q \subset M$ are type II_1 factors with $N' \cap M = C id$ and $[M:N]$ finite we show that restrictions on the standard invariants of the elementary inclusions $N \subset P$, $N \subset Q$, $P \subset M$ and $Q \subset M$ imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto $P$ and $Q$ do not commute, then $[M:N]$ is 6 or $6 + 4\sqrt 2$. In the former case $N$ is the fixed point algebra for an outer action of $S_3$ on $M$ and the angle is $π/3$, and in the latter case the angle is $cos^{-1}(\sqrt 2-1)$ and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.
A Krishnan-Sunder subfactor $R_U ßR$ of index $k^2$ is constructed from a permutation biunitary matrix $U\in M_p(\mathbb{C})\otimes M_k(\mathbb{C})$, i.e. the entries of $U$ are either 0 or 1 and both $U$ and its block transpose are unitary. The author previously showed that every irreducible Krishnan-Sunder subfactor has an intermediate subfactor by exhibiting the associated Bisch projection. The author has also shown in a separate paper that the principal and dual graphs of the intermediate subfactor are the same as those of the subfactor $R^{\grp} ßR^{H}$, where $Hß\grp$ is an inclusion of …
We consider noncommuting pairs P,Q of intermediate subfactors of an irreducible, finite-index inclusion N in M of II_1 factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value [P:N]=4cos^2(pi/2n) but none for the odd values [P:N]=4cos^2 (pi/(2n+1)). We also classify the angle values which occur between pairs of intermediate subfactors with small index over their intersection: if [P:N] < 4, then the unique nontrivial angle value is always cos^-1 (1/([P…
In this paper we present a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. Motivated by this conjecture, we determine all intermediate subfactors of Goodman-Harpe-Jones subfactors, and as a result we verify that Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a negative answer to a question motivated by a conjecture of Aschbacher-Guralnick.
We introduce a new notion of angle between intermediate subfactors and prove various interesting properties of the angle and relate it with the Jones' index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo pu…
Given a quadruple of finite index subfactors we explicitly compute the Pimsner-Popa probabilistic constant for the pair of intermediate subfactors and relate it with the corresponding Connes-Størmer relative entropy between them. This generalizes an old result of Pimsner and Popa.
Motivated by our subfactor generalization of Wall's conjecture, in this paper we determine all intermediate subfactors for conformal subnets corresponding to four infinite series of conformal inclusions, and as a consequence we verify that these series of subfactors verify our conjecture. Our results can be stated in the framework of Vertex Operator Algebras. We also verify our conjecture for Jones-Wassermann subfactors from representations of Loop groups extending our earlier results.
To any complex Hadamard matrix H one associates a spin model commuting square, and therefore a hyperfinite subfactor. The standard invariant of this subfactor captures certain "group-like" symmetries of H. To gain some insight, we compute the first few relative commutants of such subfactors for Hadamard matrices of small dimensions. Also, we show that subfactors arising from Dita type matrices have intermediate subfactors, and thus their standard invariants have some extra structure besides the Jones projections.
Let $N \subset M$ be an irreducible inclusion of type type II$_1$ factors with finite Jones index. We shall introduce the notion of normality for intermediate subfactors of the inclusion $N \subset M$. If the depth of $N \subset M$ is 2, then an intermediate subfactor $K$ for $N \subset M$ is normal in $ N \subset M$ if and only if the depths of $N \subset K$ and $K \subset M$ are both 2. In particular, if $M$ is the crossed product $N \rtimes G$ of a finite group $G$, then $K = N \rtimes H$ is normal in $N \subset M$ if and only if $H$ is a normal subgroup of $G$.
We prove that the very simple lattices which consist of a largest, a smallest and $2n$ pairwise incomparable elements where $n$ is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite depth. Using the same techniques, we give a necessary and sufficient condition for subfactors coming from Loop groups of type $A$ at generic levels to be maximal.
We describe the subfactor planar algebra of an intermediate subfactor $N\subset Q \subset M$ of an extremal subfactor $N\subset M$ of finite Jones index which is not necessarily irreducible.
In this paper, we explicitly work out the subfactor planar algebra $P^{(N \subset Q)}$ for an intermediate subfactor $N \subset Q \subset M$ of an irreducible subfactor $N \subset M$ of finite index. We do this in terms of the subfactor planar algebra $P^{(N \subset M)}$ by showing that if $T$ is any planar tangle, the associated operator $Z^{(N \subset Q)}_T$ can be read off from $Z^{(N \subset M)}_T$ by a formula involving the so-called {\em biprojection} corresponding to the intermediate subfactor $N \subset Q \subset M$ and a scalar $α(T)$ carefully chosen so as to ensure that the formula …
We study actions of locally compact groups on von Neumann factors and the associated crossed-product von Neumann algebras. In the setting of totally disconnected groups we provide sufficient conditions on an action $G\curvearrowright Q$ ensuring that the inclusion $Q \subset Q \rtimes G$ is irreducible and that every intermediate subfactor is of the form $Q \rtimes H$ for a closed subgroup $H<G$. This partially generalizes a result of Izumi-Longo-Popa [ILP98] and Choda [Ch78]. We moreover show that one can not hope to use their strategy for non-discrete groups.
A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of "type I", i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine "parent" coupling matrices Z^\pm of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z^+=Z^-, then Z i…
In this paper we discuss a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. We explore special cases of this conjecture and present supporting evidence. In particular we prove special cases of this conjecture related to some finite dimensional Kac Algebras of Izumi-Kosaki type which include relative version of Wall's conjecture for solvable groups.
Given an irreducible local conformal net A of von Neumann algebras on the circle and a finite-index conformal subnet B of A, we show that A is completely rational iff B is completely rational. In particular this extends a result of F. Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in [KLM] hold. Our proofs are based on an analysis of the net inclusion B in A; among other things we show that, for a fixed interval I, every von Neumann algebra R intermediate between B(I) and A(I) comes from an in…
Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index 3+\sqrt{5}. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and 3+\sqrt{5}. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. W…
A quadrilateral of factors is an irreducible inclusion of factors $N \subset M$ with intermediate subfactors $P$ and $Q$ such that $P$ and $Q$ generate $M$ and the intersection of $P$ and $Q$ is $N$. We investigate the structure of a non-commuting quadrilateral of factors with all the elementary inclusions $P\subset M$, $Q\subset M$, $N\subset P$, and $N\subset Q$ 2-supertransitive. In particular we classify such quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between $P$ and $Q$ for quadrilaterals coming from $α$-induction and a…
In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling matrix comparing two kinds of braided sector induction ("alpha-induction"). It has non-negative integer entries, is normalized and commutes with the S- and T-matrices arising from the braiding. Thus it is a physical modular invariant in the usual sense of rational conformal field theory. The algebraic treatment of conformal field theory models, e.g. $SU(n)_k$ m…
