# Questions tagged [braided-tensor-categories]

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140
questions

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### Why does the definition of a braided monoidal category not mention the braid equation?

Let $\mathcal{M}$ be a braided monoidal category (BMC) with braiding $\gamma$. In the definition of a BMC $\gamma$ is required to satisfy the two hexagon identities. However since "braided" ...

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### $G$-crossed (braided) fusion categories and Tannaka duality

Many important concepts in tensor category theory have their counterpart in Hopf algebra theory under Tannaka duality. They have the general form: let $A$ be an XX-algebra, and let Rep$(A)$ denote the ...

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### Does there exist a nontrivial triangular weak Hopf algebra?

Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair
($H,\mathcal{R}$) where $H$ is a WHA and
$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...

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### Is there an integral fusion category of the Ising type?

In [EGNO, Section 8.27.3], we read:
Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropic
category (namely, the core of ${\mathcal C}$) using ﬁnite groups (via the
...

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### Is there an alternating power functor on braided monoidal categories?

Let $\mathsf{C}$ be a braided monoidal $\mathbb{C}$-linear category and $V\in \mathsf{C}$ any object. Then we can form the tensor power $V^{\otimes 2}$.
The braid group $B_2$ acts on $V^{\otimes 2}$, ...

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### Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?

Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined as
$$...

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1
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### Is there a non-split super-modular positive integral fusion category?

We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...

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### Braidings on Temperley-Lieb Category

Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes ...

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### Is anything known about the center of the Fomin-Kirillov algebra?

Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...

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### Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?

In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the ...

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### Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?

Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...

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### Software for working with fusion categories

One way to describe fusion categories is via a fusion system: several lists of numbers that define the fusion ring, associator, braiding (if it exists), etc. Often, these sets of numbers are quite big,...

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### Yetter-Drinfeld modules for Hopf monads

1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...

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### Reshetikhin-Turaev invariants from extended 3d TQFTs

Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants
$$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$
for ...

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### Question about references for proof of Proposition 1.3 in P. Deligne & J.S. Milne's article on "Tannakian Categories"

I am trying to understand the proof of proposition 1.3 in the following article by P. Deligne & J.S. Milne: Tannakian Categories.
They reference Saavedra Rivano's 1972 "Catégories ...

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### Does a fusion ring with F and R-symbols uniquely determine a braided tensor category?

Background : In mathematical physics, 'anyons' in (2+1) dimensional systems are described by braided tensor categories. The anyon types correspond to the irreducible objects of the category. From such ...

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### Tannakian reconstruction for braided categories

Let $\mathcal{C}$ be a symmetric monoidal category. One can imagine a theorem
Tannakian reconstruction: If $\mathcal{B}$ is a braided monoidal category and $F:\mathcal{B}\to \mathcal{C}$ is a functor ...

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### Is a Lagrangian subgroup of a metric group isomorphic to its quotient?

A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...

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### Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?

What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...

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### Naturality of ribbon category twists

Tortile Tensor Categories by Shum defines a twist to be a natural transformation $\theta : \operatorname{Id} \to \operatorname{Id}$ satisfying some axioms. However, wikipedia and nLab worded the ...

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### Is there a notion of "knot category"?

Consider a rigid braided monoidal category, with braiding $\beta_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon_x : 1 \to a \otimes a^*, \bar\epsilon_x : a^* \...

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### Lagrangian subcategories of (non-pointed) braided tensor categories

I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...

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### Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors?

This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular&...

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2
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### Where does the univeral $R$-matrix of $U_q(\mathfrak g)$ live?

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_q(\mathfrak g)$ denote the Drinfeld-Jimbo quantum group associated to $\mathfrak g$. I will assume that $U_q(\mathfrak g)$ is a $\mathbb C(...

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### Motivating quantum groups from knot invariants

Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...

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### Coherence theorem in braided monoidal categories

In MacLane's Categories for the working mathematician, the author shows that the evaluation at 1 gives an equivalence of categories $\mathrm{hom}_{\mathrm{BMC}}(B,M)\simeq M_0$ where $B$ is the braid ...

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### How to calculate the Lagrangian subgroup of $G\oplus\hat{G}$?

Let $G$ be an finite abelian group. We have known the following things:
Denote the Drinfeld center of $\operatorname{Rep}(G)$ by $\mathfrak{Z}_1(\operatorname{Rep}...

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1
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### Braided R-matrices for finite action groupoids

1. Algebra from action groupoids
Let $G$ be a finite group acting on a finite set $X$ from the
right (denoted in element as $x^{g}$). We have an algebra (of the
action groupoid) over $\mathbb{C}$: the ...

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2
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### What is the proof of the compatibility of a braiding with the unitors?

I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$,
$$
\lambda_A \circ c_{A,I}=\rho_{A},
$$
for any ...

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### Why is 'every braided monoidal category spacial'? [duplicate]

In his 2009 survey, Selinger ("A survey of graphical languages for monoidal categories") defines the notion of a 'spacial monoidal category', which (in his graphical calculus) allows one to ...

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### An introductory reference for tensor networks

I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?

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### Connection between braided tensor categories and local systems on moduli of stable marked genus zero curves

I'm looking for references regarding an unpublished Deligne's manuscript "Une descrption de catégorie tressée (inspiré par Drinfeld)" and the subject it touches, that is described in the ...

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### Braided monoidal category, example

Let $H$ be a cocommutative hopf algebra.
Let $M$ be the category of $H$-bimodules.
Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?

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### Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$

$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...

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2
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### Constructing the inverse of a braiding in a braided pivotal category

Assume we have a braided pivotal monoidal category. This means we assume the braiding $c$ to be a natural isomorphism. But looking at the corresponding string diagram, it seems to me as if we could ...

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### Drinfeld center of $\mathrm{Mod}_R$

Let $R$ be a commutative ring and let $\mathrm{Mod}_R$ be the category of (left) $R$-modules.
Question: Is it true that the categories $\mathcal{Z}(\mathrm{Mod}_R)$ and $\mathrm{Mod}_R$ are ...

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0
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### Recovering the center of a monoid from the Drinfeld center

The Drinfeld center construction is intended to be a categorification of the center of a monoid. It seems to be folklore (eg this answer or this one) that when the Drinfeld center is taken over a ...

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### Ordered logic is the internal language of which class of categories?

Wikipedia says:
The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system.
"A Fibrational Framework for Substructural and Modal ...

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### Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?

Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...

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### Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true
($R$-Mod) $\simeq$ ($S$-Mod).
$S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...

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### Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...

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### Comodule Morita equivalence for Hopf algebras

Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\...

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### Is there a non-degenerate quadratic form on every finite abelian group?

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...

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### Does the functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ have adjoints?

Let $\mathcal{C}$ be a braided monoidal category. We have a canonical functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ from $\mathcal{C}$ to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ sending an ...

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### When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?

Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{...

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### On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...

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### Representation theory in braided monoidal categories

The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ...

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### additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...

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### Braided category inside braided 2-category

Let $\mathcal{C}$ be a semistrict braided monoidal $2$-category in the sense of [BN] (so in particular a strict $2$-category). Let $\mathcal{C}_1$ be the category of $1$-morphisms (objects) and $2$-...

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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 $Y$ ...