We construct the quantum s-tuple subfactors for an AFD II_1 subfactor with finite index and depth, for an arbitrary natural number s. This is a generalization of the quantum multiple subfactors by J.Erlijman and H.Wenzl, which generalizes the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.
We associate Popa systems (= standard invariants of subfactors) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be ``represented'' on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to …
We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the `"quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equiva…
We give a precise definition for when a subfactor arises from a conformal net which can be motivated by classification of defects. We show that a subfactor $N \subset M$ arises from a conformal net if there is a conformal net whose representation category is the quantum double of $N\subset M$.
Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T).
We construct inclusions of the form $(B_0\otimes P)^G\subset (B_1\otimes P)^G$, where $G$ is a compact quantum group of Kac type acting on an inclusion of finite dimensional $\c^*$-algebras $B_0\subset B_1$ and on a $II_1$ factor $P$. Under suitable assumptions on the actions of $G$, this is a subfactor, whose Jones to er and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spi…
We define generalised notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.
Subfactor standard invariants encode quantum symmetries. The small index subfactor classification program has been a rich source of interesting quantum symmetries. We give the complete classification of subfactor standard invariants to index $5\frac{1}{4}$, which includes $3+\sqrt{5}$, the first interesting composite index.
We construct a certain `cobordism category' ${\cal D}$ whose morphisms are suitably decorated cobordism classes between similarly decorated closed oriented 1-manifolds, and show that there is essentially a bijection between (1+1-dimensional) unitary topological quantum field theories (TQFTs) defined on ${\cal D}$, on the one hand, and Jones' subfactor planar algebras, on the other.
For a finite-index $\mathrm{II}_1$ subfactor $N \subset M$, we prove the existence of a universal Hopf $\ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf $\ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by …
In this paper, we introduce Frobenius von Neumann algebras and study quantum convolution inequalities. In this framework, we unify quantum Young's inequality on quantum symmetries such as subfactors, and fusion bi-algebras studied in quantum Fourier analysis. Moreover, we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case. We also prove quantum smooth entropic convolution inequalities. We obtain the positivity of comultiplications of subfactor planar algebras, which is stronger than the quantum Schur product theorem. All these inequalities pr…
We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.
By a construction of Vaughan Jones, the bipartite graph $Γ(A)$ associated with the natural inclusion of $\mathbb C$ inside a finite-dimensional $C^*$-algebra $A$ gives rise to a planar algebra $\mathcal P^{Γ(A)}$. We prove that every subfactor planar subalgebra of $\mathcal P^{Γ(A)}$ is the fixed point planar algebra of a uniquely determined action of a compact quantum group $\mathbb G$ on $A$. We use this result to introduce a conceptual framework for the free wreath product operation on compact quantum groups in the language of planar algebras/standard invariants of subfactors. Our approach …
To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_μU(2). The deformation parameter is determined by -1<μ<0 and [M:N]=|μ+μ^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_μU(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. μis negative as M is a real bimodule.
We study (dual) Longo-Rehren subfactors $M\otimes M^{opp} \subset R$ arising from various systems of endomorphisms of M obtained from alpha-induction for some braided subfactor $N\subset M$. Our analysis provides useful tools to determine the systems of R-R morphisms associated with such Longo-Rehren subfactors, which constitute the ``quantum double'' systems in an appropriate sense. The key to our analysis is that alpha-induction produces half-braidings in the sense of Izumi, so that his general theory can be applied. Nevertheless, alpha-induced systems are in general not braided, and thus ou…
It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus …
In this paper, we establish the general theory of (2+1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev-Viro-Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin-Turaev invariant constructed from a tube system,…
Given a finite-index and finite-depth subfactor, we define the notion of \textit{quantum double inclusion} - a certain unital inclusion of von Neumann algebras constructed from the given subfactor - which is closely related to that of Ocneanu's asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor $R^H \subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset …
We call a subfactor trivial if it is isomorphic with the obvious inclusion of N into matrices over N. We prove the existence of type II_1 factors M without non-trivial finite index subfactors. Equivalently, every M-M-bimodule with finite coupling constant, both as a left and as a right M-module, is a multiple of L^2(M). Our results rely on the recent work of Ioana, Peterson and Popa, who proved the existence of type II_1 factors without outer automorphisms.
A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with gho…
We prove existence of subfactors of finite depth of the hyperfinite II_1 factor with indices (5+sqrt{13})/2= 4.302... and (5+sqrt{17})/2=4.561.... The existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then. These are the only known subfactors with finite depth which do not arise from classical groups, quantum groups or rational conformal field theory.
Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of…
Discrete subfactors include a particular class of infinite index subfactors and all finite index ones. A discrete subfactor is called local when it is braided and it fulfills a commutativity condition motivated by the study of inclusion of Quantum Field Theories in the algebraic Haag-Kastler setting. In [BDG21], we proved that every irreducible local discrete subfactor arises as the fixed point subfactor under the action of a canonical compact hypergroup. In this work, we prove a Galois correspondence between intermediate von Neumann algebras and closed subhypergroups, and we study the subfact…
In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m$ times}} : m \geq 1 \}$. For each $…
We are concerned with the center (=quantum double) of tensor categories and prove generalizations of several results proven previously for quantum doubles of Hopf algebras. We consider F-linear tensor categories C with simple unit and finitely many isomorphism classes of simple objects. We assume that C is either a *-category (i.e. there is a positive *-operation on the morphisms) or semisimple and spherical over an algebraically closed field F. In the latter case we assume that dim C=sum_i d(X_i)^2 is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove:…
We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.
In this article, we will prove that the subsectors of $α$-induced sectors for $M \rtimes \hat{G} \supset M$ forms a modular category, where $M \rtimes \hat{G}$ is the crossed product of $M$ by the group dual $\hat{G}$ of a finite group $G$. In fact, we will prove that it is equivalent to Müger's crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of $Δ$(not necessarily non-degenerate) obtained from a Longo-Rehren inclusion $A \supset B_Δ$ under certain assumptions. We will apply the above description of the quantum double of $Δ$ to the Reshet…
The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data DHg fits into a family $D^ωHg_{2n+1}$, where $n\ge 0$ and $ω\in \bbZ_{2n+1}$. We show $D^0 Hg_{2n+1}$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $Z_{2n+1}$. Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of I…
{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^{p}$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outlin…
Equivalence of states and bimodule quantum channels
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；School of Mathematical Sciences, Hebei Normal University（河北师范大学数学科学学院）；Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University（清华大学丘成桐数学科学中心及数学系）
Let $\mathbb{G}$ be a free (unitary or orthogonal) quantum group. We prove that for any non-amenable subfactor $N\subset L^\infty(\mathbb{G})$, which is an image of a faithful normal conditional expectation, and for any $σ$-finite factor $B$, the tensor product $N \otimes B$ has no Cartan subalgebras. This generalizes our previous work that provides the same result when $B$ is finite. In the proof, we establish Ozawa--Popa and Popa--Vaes's weakly compact action on the continuous core of $N \otimes B$ as the one relative to B, by using an operator valued weight to B and the central weak amenabi…
