arXiv subjects
Publications and source records attributed to David Penneys.
In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.
We show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although this result could be proved as a corollary of our previous article on realizations of algebra objects and discrete subfactors, we prove it here directly via categorical methods without passing through subfactor theory.
We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal V \to Z(\mathcal T)$. We would like to understand this further; in a future paper we show that the functor is strong if and only if the enriched category is `complete' in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor $\mathsf{Rep}(G) \to Z(\mathcal T)$ for some finite group $G$ and a monoidal category $\mathcal T$, and produces a new monoidal category $\mathcal T // G$. In our setting, given any braided oplax monoidal functor $\mathcal V \to Z(\mathcal T)$, for any braided $\mathcal V$, we produce $\mathcal T // \mathcal V$: this is not usually an `honest' monoidal category, but is instead $\mathcal V$-enriched. If $\mathcal V$ has a braided lax monoidal functor to $\mathsf{Vec}$, we can use this to reduce the enrichment to $\mathsf{Vec}$, and this recovers de-equivariantization as a special case.
Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.
In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{C}$. A C*/W*-algebra object in $\mathcal{C}$ is an algebra object $\mathbf{A}$ in $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ is a $\mathcal{C}$-module C*/W*-category respectively. When $\mathcal{C}={\sf Hilb_{f.d.}}$, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C*-algebra objects in $\mathcal{C}$ and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra $\mathbf{M}$ in $\mathcal{C}$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.
We generalize Jones' planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional `anchor lines' which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when $\mathcal{C}$ is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$ and pivotal module tensor categories over $\mathcal{C}$ equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.
Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $\mathrm{Tr}_{\mathcal C}$ comes equipped with natural isomorphisms $τ_{x,y}:\mathrm{Tr}_{\mathcal C}(x \otimes y) \to \mathrm{Tr}_{\mathcal C}(y \otimes x)$, which we call the traciators. This situation lends itself to a diagramatic calculus of `strings on cylinders', where the traciator corresponds to wrapping a string around the back of a cylinder. We show that $\mathrm{Tr}_{\mathcal C}$ in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects $A$ and $B$, we prove that $\mathrm{Tr}_{\mathcal C}(A)$ and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are again algebra objects. Moreover, provided certain mild assumptions are satisfied, $\mathrm{Tr}_{\mathcal C}(A)$ and $\mathrm{Tr}_{\mathcal C}(A \otimes B)$ are semisimple whenever $A$ and $B$ are semisimple.
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 state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the $3^G$ subfactor planar algebras. We prove the conjecture in the case when $|G|$ is odd. To do so, we find an action of $G$ on the reduced subfactor planar algebra at $f^{(2)}$, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.
Given a II$_1$-subfactor $A\subset B$ of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as $A-A$ bimodule maps among the bimodules $\bigotimes_A^n L^2(B)$. As a corollary, we get a lower bound on the dimension of the centralizer algebras $A_0'\cap A_{2n}$ for infinite index subfactors, and we also get that $A_0'\cap A_{2n}$ is nonabelian for $n\geq 2$, where $(A_n)_{n\geq 0}$ is the Jones tower for $A_0=A\subset B=A_1$. We also show that the annular GICAR/planar rook category acts as maps amongst the $A$-central vectors in $\bigotimes_A^n L^2(B)$, although this action may be degenerate. We prove these results in more generality using bimodules. The embedding of the GICAR category builds on work of Connes and Evans who originally found GICAR algebras inside Temperley-Lieb algebras with finite modulus.
This article proves the existence and uniqueness of a subfactor planar algebra with principal graph consisting of a diamond with arms of length 2 at opposite sides, which we call 2D2. We also prove the uniqueness of the subfactor planar algebra with principal graph 4442. We conjecture this will complete the list of subfactor planar algebras at index $3+\sqrt{5}$.
We give the classification of subfactor planar algebras at index exactly 5. All the examples arise as standard invariants of subgroup subfactors. Some of the requisite uniqueness results come from work of Izumi in preparation. The non-existence results build upon the classification of subfactor planar algebras with index less than 5, with some additional analysis of special cases.
From a planar algebra, we give a functorial construction to produce numerous associated $C^*$-algebras. Our main construction is a Hilbert $C^*$-bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular $C^*$-algebras. By compressing this system, we obtain various canonical $C^*$-algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-)Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical $C^*$-algebras associated to a planar algebra.
We study the $C^*$-algebras arising in the construction of Guionnet-Jones-Shlyakhtenko (GJS) for a planar algebra. In particular, we show they are pairwise strongly Morita equivalent, we compute their $K$-groups, and we prove many properties, such as simplicity, unique trace, and stable rank 1. Interestingly, we see a $K$-theoretic obstruction to the GJS $C^*$-algebra analog of Goldman-type theorems for II$_1$-subfactors. This is the second article in a series studying canonical $C^*$-algebras associated to a planar algebra.
We classify irreducible II_1 subfactors A \subset B such that B \ominus A is reducible as an A-A bimodule, with index at most 6+1/5, leaving aside the composite subfactors at index exactly 6. Previous work has already achieved this up to index 3+\sqrt{5} \approx 5.23. We find there are exactly three such subfactors with index in (3+\sqrt{5}, 6+1/5], all with index 3+2\sqrt{2}. One of these comes from SO(3)_q at a root of unity, while the other two appear to be closely related, and are `braided up to a sign'.
Given a pair of fusion categories $C$ and $D$, we may form the free product $C * D$ and the tensor product $C \boxtimes D$. It is natural to think of the tensor product as a quotient of the free product. What other quotients are possible? When $C=D=A_2$, there is an infinite family of quotients interpolating between the free product and the tensor product (closely related to the $A_{2n-1}^{(1)}$ and $D_{n+2}^{(1)}$ subfactors at index 4). Bisch and Haagerup discovered one example of such an intermediate quotient when $C=A_2$ and $D=T_2$, and suggested that there might be another family here. We show that such quotients are characterized by parameters $n \geq 1$ and $ω$ with $ω^{2n}=1$. For $n=1,2,3$, we show $ω$ must be 1, and construct the corresponding quotient ($n=1$ is the tensor product, $n=2$ is the example discovered by Bisch and Haagerup, and $n=3$ is new). We further show that there are no such quotients for $4 \leq n \leq 10$. Our methods also apply to the case when $C=D=T_2$, and we prove similar results there. During the preparation of this manuscript we learnt of an independent result of Liu's on subfactors. With the translation between the subfactor and fusion category settings provided here, it follows there are no such quotients for any $n \geq 4$.
We construct subfactors where one of the principal graphs is a spoke graph using an algorithm which computes two-strand jellyfish relations. One of the subfactors we construct is a 3^{Z/4} subfactor known to Izumi, which has not previously appeared in the literature. To do so, we provide a systematic treatment of the space of second annular consequences, which is analogous to Jones' treatment of the space of first annular consequences in his quadratic tangles article. This article is the natural followup to two recent articles on spoke subfactor planar algebras and the jellyfish algorithm. Work of Bigelow-Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison-Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish.
Determining which bipartite graphs can be principal graphs of subfactors is an important and difficult question in subfactor theory. Using only planar algebra techniques, we prove a triple point obstruction which generalizes all known initial triple point obstructions to possible principal graphs. We also prove a similar quadruple point obstruction with the same technique. Using our obstructions, we eliminate some infinite families of possible principal graphs with initial triple and quadruple points which were a major hurdle in extending subfactor classification results above index 5.