Questions tagged [braided-tensor-categories]
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125
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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}...
- 31
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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 ...
- 1,432
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About symmetry, braids, and pseudo-functors.
Let $(\mathcal{C}, \otimes, I)$ a monoidal category, and let $\mathbb{B}(\mathcal{C})$ the bicategory (with only one object and $(\mathcal{C}, \otimes, I)$ as (monoidal) category of morphisms and ...
- 4,095
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Braidings and Isomorphism Classes in a Monoidal Category
Let $X$ be an object in a monoidal category $({\cal C}, \otimes)$, and $\gamma:X \otimes X \to X \otimes X$ a braiding (that is to say a morphism in ${\cal C}$ from $X \otimes X$ to itself that ...
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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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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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2
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1
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443
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Graphical calculus in braided G crossed fusion categories: Explanation request and a question
I am trying to understand the equivalence between the 2 category of braided G crossed categories and the 2 category of braided categories containing Rep(G) as a symmetric category. The references in ...
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Is there a notion of partial trace in a ribbon category?
I've seen some definitions of "right partial trace" and "left partial trace" in http://arxiv.org/abs/1103.1660, but these don't seem canonical in any way.
The motivation for this questions is that I'...
- 21
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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. ...
- 5,485
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When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?
I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...
- 5,485
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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 $\...
- 1,104
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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 ...
- 89
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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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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_{\...
- 974
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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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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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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 ...
- 4,760
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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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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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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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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$-...
- 1,527
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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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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 ...
- 6,041
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Braidings for Comodules of Co-quasi-triangular Hopf algebra
Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a ...
- 1,429
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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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