Questions tagged [fusion-categories]
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Why are fusion categories interesting?
In the same vein as Kate and Scott's questions, why are fusion categories interesting? I know that given a "suitably nice" fusion category (which probably means adding adjectives such as "unitary," "...
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Are there two groups which are categorically Morita equivalent but only one of which is simple
Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) ...
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Sum of squares and divisibility
Consider an integer of the form $$N = 1 + \sum_{i=1}^r d_i^2$$ where $d_i \in \mathbb{N}_{\ge 3}$ and $d_i^2$ divides $N$.
Question: Must $r$ be greater than or equal to $9$?
Checking (with SageMath): ...
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Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?
The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
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Non weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$ below)?
$$\scriptsize{\begin{...
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Unitary structures on fusion categories
A unitary fusion category is a fusion category with a $C^*$-tensor structure.
Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more ...
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How do I describe a fusion category given a subfactor?
I felt like following up on Kate's question. There were some good motivational answers there.
Given a pair of factors M < N, there is a standard way to construct a 2-category whose objects are M ...
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Are there interesting semisimple algebras in non-semisimple categories?
Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple ...
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A cohomology theory for fusion categories
It is well known that for a finite group $G$, the associator of the fusion category of $G$-graded $k$-vector spaces is given by an element of $H^3(G,k^*)$, up to equivalence of categories. ($k^*$ is ...
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Does every Frobenius algebra in a monoidal *-category give a Q-system?
Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...
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A linear category with objects of infinite length but which is otherwise finite?
Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...
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Is there a "killing" lemma for G-crossed braided fusion categories?
Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.
Premodular categories
In braided ...
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What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?
For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one ...
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Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...
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Symmetries of module categories over the category of representations of quantum $sl(2)$
The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...
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Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects
Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...
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Is there a subfactor construction involving 2-groups?
I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an ...
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Is there a fusion category with an object which does not commute with its dual?
Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)?
Feel free to add adjectives such as pivotal, spherical, ...
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What is a true invariant of $G$-crossed braided fusion categories?
Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences.
(Spherical) fusion categories have ...
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Is there a fusion category not Grothendieck equivalent to a unitary one?
We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) for the notion of (unitary) fusion category. Two fusion categories are Grothendieck equivalent if they have the ...
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Suppose C and D are Morita equivalent fusion categories, can you say anything about R I: C->Z(C)=Z(D)->D?
If C and D are (higher) Morita equivalent fusion categories, then the Drinfel'd centers Z(C) and Z(D) are braided equivalent. Given any fusion category C we have a restriction functor Z(C)->C (by ...
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Is the modularisation of a unitary fusion category always unitary?
Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
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Are there workable algebraic geometry approaches for the pentagon equation?
A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer ...
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What are the intermediate subfactors of the tensor product of two maximal subfactors?
Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
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Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation
In this post, irrep and dim mean "irreducible complex representation" and "dimension", respectively.
It would be helpful (in a problem of monoidal category) to find a finite group $G$ with (at least) ...
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Twists, balances, and ribbons in pivotal braided tensor categories
Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures ...
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How does the Tannaka duality work for weak Hopf algebras and fusion categories?
I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
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Brauer-Picard for a fusion category coming from a quantum group
In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...
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How weird can Modular Tensor Categories be over non-algebraically closed fields?
I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?
The reason I am interested in this is that my ...
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If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
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Conditions on the fusion data of symmetric fusion category
We know that every symmetric fusion category (SFC) gives rise to data
$N^{ij}_k$ that describe the fusion of simple objects:
$i\times j = N^{ij}_k k$, and the data $\theta_i =\pm 1$ that describe the ...
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realizing fusion categories as subfactors of the hyperfinite
Let R be the hyperfinite II_1 or the hyperfinite III_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules.
This question is inspired by the recent ...
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R-matrices and symmetric fusion categories
Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ ...
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Is there a source for a diagrammatic description of the induction functor C->Z(C)?
Suppose that C is a fusion category (over the complex numbers) and that Z(C) is its Drinfel'd center. By definition an object in Z(C) consists of an object V in C together with a collection of half-...
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Morita equivalent algebras in a fusion category
Let $\mathcal{C}$ be a braided $\mathbb{k}$-linear fusion category ($\mathbb{k}$ algebraically closed; if necessary to answer my question you can also assume $\mathcal{C}$ to be pivotal or even ...
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What classifies involutive automorphisms on finite groups? What classifies involutions on finite based rings?
Groups
Let $G$ be a finite group. An involutive automorphism on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.
Question 1. What classifies involutive automorphisms on a given (non-...
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How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?
In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").
Physically this modular fusion category describes the ...
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Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?
The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...
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Is there an integral fusion ring which is not of Frobenius type?
Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...
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Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes?
Let $C$ be a fusion category with simple objects $X_1,...,X_n $, and let $Y_1,...,Y_n$ be objects with each $Y_i$ isomorphic to $X_i$. Is there a monoidal auto-equivalence $F:C \rightarrow C $ which ...
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6j symbols of SU(4) at level 4
Does anybody know of a reference that gives the (quantum) 6j symbols of SU(4) at level 4?
Alternatively, I know the S-matrix and the fusion rules, in the form
$a \times b = \sum_i N^{ab}_{c_i} c_i$
...
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Fusion category and Hopf algebra
Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed ...
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Fusion Classification of $U_q(sl_N)$ Categories
Frohlich and Kerler classify categories with $SU(2)_k$ fusion rules and Kazdhan-Wenzl expand this to $SU(N)_k$ categories. In both cases, unless I am missing something, the classification are ...
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Does the pentagon axiom force the associativity constraint to be a natural isomorphism?
Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
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How to translate connection on four graphs to quantum 6j symbols
I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...
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Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
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An alternative Cauchy theorem on Hopf algebras
Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].
We are interesting in an alternative ...
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Is there a non-trivial Hopf algebra without left coideal subalgebra?
Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\...
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Drinfeld center of a Deligne tensor product
Let $\mathcal{C}$ and $\mathcal{D}$ be two tensor categories (if necessary, assume they are fusion categories). Is the canonical braided monoidal functor $$\mathcal{Z}(\mathcal{C})\boxtimes\mathcal{Z}(...
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Is there a strongly noncommutative fusion category?
A fusion category is called noncommutative if its Grothendieck ring is noncommutative. Let us call a fusion category strongly noncommutative if every fusion category Morita equivalent to it (i.e. same ...
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