Questions tagged [braided-tensor-categories]

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Integrals and finite dimensionality in braided Hopf algebras

Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$. A nonzero left integral in $A$ is a nonzero ...
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Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian? [stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)] In ...
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1answer
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Is the category $\operatorname{sVect}$ an “algebraic closure” of $\operatorname{Vect}$?

$\DeclareMathOperator\sVect{sVect}\DeclareMathOperator\Vect{Vect}$The category $\sVect_k$ of (let's say finite-dimensional) super vector spaces can be obtained from the category $\Vect_k$ of (finite-...
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On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
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Tensor algebras in the bicategory $\mathsf{2Vect}$

To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both ...
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1answer
103 views

Examples of strict monoidal categories and monoidal categories with nontrivial associators

What are some "natural" motivating examples of the following: i) A strict monoidal category, ii) A monoidal with non-trivial associatots? For i) the only examples I know are categories which ...
5
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1answer
123 views

Nonbraided rigid monoidal category where left and right duals coincide

In a braided rigid monoidal category $(\mathcal{M},\otimes)$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no ...
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Integrals in noncommutative graded algebras which are not necessarily Hopf

Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
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Categorical construction of comodule category of FRT algebra

Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be ...
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830 views

The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
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Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$?

Deligne's theorem on tensor categories states that for any symmetric tensor category $\mathcal{C}$ satisfying the subexponential growth condition, there is a fiber functor to $\mathsf{sVec}$ and that $...
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Scaling Yetter--Drinfeld Modules

A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
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semisimplicity of maps in braided vector spaces

Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$. This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
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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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What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
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1answer
131 views

Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked. I know that there ...
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2answers
318 views

Enrichments vs Internal homs

Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor $$ X \otimes -: \cal{C} \to \cal{C}, $$ for ...
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4answers
532 views

The dual of a dual in a rigid tensor category

For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
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702 views

Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?

The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...
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Non-semisimple representations of the braid group in a semisimple braided category

Suppose $\mathcal{C}$ is a semisimple braided tensor category (over $\mathbb{C}$, with finite dimensional hom spaces) and $X$ an object in $\mathcal{C}$. Then for each n > 0 the braid group $B_n$ ...
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1answer
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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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Question about terminology, and reference request related to the braid operad

Let $\Delta_n$ stand for the Garside element of the braid group $B_n$. It turns out that the family of all Garside elements have the following ``operadic'' property: $$ \Delta_n\left[ \Delta_{k_1},\...
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2answers
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Uniqueness of character for Z_+-rings

I have a question about the proof of proposition $3.3.6(3)$ in "Tensor Categories" by Etingof et al.. This part states that for $A$, transitive unital $\mathbb Z_+$-ring, there is a unique character ...
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How does the relative Drinfeld center interact with the relative Deligne tensor product?

Let $\mathcal{C}$ be a fusion category, and $\mathcal{M}, \mathcal{N}$ semisimple $(\mathcal{C}, \mathcal{C})$-bimodule categories. The left $\mathcal{C}$-action is denoted as $- \triangleright - \...
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1answer
406 views

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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1answer
265 views

What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?

Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
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Braid groups representations on infinite dimensional vector spaces

Let $V$ be an infinite dimensional complex vector space. Let $R:V\otimes V\to V\otimes V$ be a solution to the quantum Yang Baxter Equation. In other words: $R$ is invertible and satisfies the ...
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1answer
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Bimodule categories realized as internal bimodules

Let $\mathcal C$ be a finite tensor category, and $\mathcal M$ a finite left $\mathcal C$-module category. By a result of P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik (http://www-math.mit.edu/~...
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318 views

About a categorical definition of graded (coloured) algebra

The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not ...
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1answer
136 views

Balanced monoidal and homotopy symmetric

It's probably a very simple question but I am not sure about the reference. In the definition of a balanced monoidal category we require that the braiding isomorphims $$c_{V, W}: V \otimes W \to W \...
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Which Drinfeld centers are balanced monoidal, i.e. have a twist?

A twist is an automorphism $\theta$ of the identity functor of a monoidal category with braiding $c$, such that $\theta_{X \otimes Y} = c_{Y,X} c_{X,Y} (\theta_X \otimes \theta_Y)$. A braided monoidal ...
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1answer
175 views

On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category. Question: What about the converse, i.e., can we characterize every unitary modular tensor ...
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1answer
214 views

representation of a group and its center

(I asked the following question at StackExchange but received no answer.) Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...
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Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
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1answer
273 views

Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface?

In Huang & Lepowsky's series of papers A theory of tensor products for module categories for a vertex operator algebra, they defined for a rational vertex algebra $V$ the $P(z)$ tensor product of ...
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References of an operator $T: V \otimes V \to V \otimes V$

Let $V$ be a vector space with a basis $v_1, \ldots, v_n$ and let $X_{ij} = v_i \otimes v_j$. Then $X_{ij}, i,j=1,\ldots, n$, is a basis of $V \otimes V$. Let $T: V \otimes V \to V \otimes V$ be the ...
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1answer
118 views

How nontrivial can “central extensions of ribbon fusion categories” be?

In a sense, this is a follow up on this question, but one PhD programme later. Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...
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221 views

When are the braid relations in a quasitriangular Hopf algebra equivalent?

Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations: $$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$ $$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
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Braided Hopf algebras and Quantum Field Theories

It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...
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1answer
287 views

When modular tensor categories are equivalent?

I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there. I would like to know when we say that two modular tensor categories are equivalent. Is it ...
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1answer
129 views

Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
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1answer
323 views

Faithful exact functors to tensor categories

Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to ...
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1answer
519 views

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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What's the relation between half-twists, star structures and bar involutions on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
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422 views

Gauss-Milgram formula for fermionic topological order?

For Bosonic topological order, a very useful formula was proved to be true: $\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $ (for more detail: $d_a$ is the quantum dimension of anyon ...
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Structure of Lagrangian algebras in the center of a fusion category

(1) Let $\mathcal F$ be a spherical fusion tensor category. Then Müger showed that $R=\bigoplus_{H\in\mathrm{Irr}(\mathcal F)} H\boxtimes H^\mathrm{op}$ canonically has the structure of a Frobenius ...
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What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
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137 views

Cyclic structure on a balanced (or ribbon) monoidal category

As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of ...
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When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

Given a category $\mathcal{C}$, we can define the category of endofunctors $\operatorname{Cat}(\mathcal{C})$, with objects functors $F: \mathcal{C} \to \mathcal{C}$ and morphisms natural ...
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421 views

Why does a tetracategory with one object, one 1-morphism and one 2-morphism give a symmetric monoidal category

According to the periodic table of k-tuply monoidal n-categories, it should be the case that a tetracategory (= weak 4-category) with one object, one 1-morphism and one 2-morphism is effectively ...