All Questions
Tagged with hopf-algebras monoidal-categories
48 questions
0
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44
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Categorical duals for Yetter-Drinfeld modules [duplicate]
Yetter-Drinfeld (YD) modules appear naturally in the theory of Hopf algebras. They are both modules and comodules at the same time, satisfying a certain compatibility condition, as presented here. The ...
3
votes
0
answers
60
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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 ...
- 727
6
votes
0
answers
122
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If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?
Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...
- 63.9k
3
votes
2
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135
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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(...
- 727
3
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0
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98
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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 $(V,\rho,\Delta)$ ...
- 764
3
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0
answers
133
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Tannaka duality for Hopf algebroids
Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...
- 764
8
votes
1
answer
311
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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 ...
- 727
9
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0
answers
326
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Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?
Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...
- 3,446
2
votes
1
answer
152
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Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
4
votes
2
answers
585
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A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?
A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
8
votes
2
answers
852
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Is a Hopf algebra a group object of some category?
The page of ncatlab on group object states that:
A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf
algebra.
Question: Is a (noncommutative) Hopf algebra a group object of some ...
18
votes
2
answers
1k
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Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
5
votes
1
answer
473
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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 $\otimes_{H}$ ?
- 71
6
votes
1
answer
591
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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 ...
- 91
5
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0
answers
257
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Derived category of an abelian monoidal category
For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...
- 203
1
vote
1
answer
210
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Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra
It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
- 1,144
0
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0
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78
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Monoid objects constructed from duals
Let $(M,\otimes)$ be a rigid monoidal category, for which left and right duals coincide. For any object $X \in M$, we can define a monoid structure on $X \otimes X^*$: Multiplication is defined by ...
- 1,144
2
votes
1
answer
218
views
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 $\...
- 1,144
2
votes
1
answer
127
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Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
- 219
3
votes
1
answer
456
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Rigidity for the category of comodules over a Hopf algebra
On this page
https://ncatlab.org/nlab/show/rigid+monoidal+category
there is a discussion of rigidity (left-right duality) for the catagory of
modules over a Hopf algebra. What happens if we look at ...
- 993
10
votes
3
answers
856
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Tannaka-Krein duality in Chari-Pressley's book
I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum ...
- 7,426
2
votes
0
answers
169
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The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories
Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces.
Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...
- 61
11
votes
4
answers
2k
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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 ...
- 835
4
votes
0
answers
103
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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 ...
- 835
4
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0
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159
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Hopf monoid from comonoidal structures
Let $\mathcal{V}$ be a closed braided monoidal category and $\mathcal{V}-Cat$ the monoidal bicategory of small $\mathcal{V}$-enriched categories. Let $\mathcal{C}$ be a pseudo-comonoid in $\mathcal{V}-...
- 1,813
8
votes
1
answer
826
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Modules over Hopf Algebras and $E_2$-algebras
Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf
I was wondering if anybody knows of a nice relationship between ...
- 527
4
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1
answer
317
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Tannaka-Krein reconstruction and rigidity
Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...
- 718
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3
answers
327
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Unbiased Hopf algebras
In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
- 43.2k
9
votes
4
answers
1k
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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.
- 993
12
votes
2
answers
624
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On the isomorphism problem of enveloping algebras
Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
- 395
10
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2
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504
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A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module
For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
- 1,161
2
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0
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163
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Quantum invariant: The canonical $2$-tensor
In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
- 395
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0
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90
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Detecting skew-primitives in representation categories
Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf ...
- 395
4
votes
0
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130
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Category of (co)commutative Hopf monoids in an exact category
I'm transferring this question over from SE, since it didn't get much attention over there.
Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...
- 3,019
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0
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217
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Categorical interpretation of quantum double $D(A,B,\eta)$
It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties:
The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ...
- 1,813
5
votes
1
answer
167
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Classification of pointed Hopf algebras up to gauge equivalence
The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. ...
- 395
4
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0
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310
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Nichols Algebras as Braided Hopf Algebras
Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
- 159
2
votes
1
answer
197
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When are Morita classes represented by certain structured algebra objects?
Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...
- 1,209
21
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2
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3k
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Does projective imply flat?
Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
- 54.6k
9
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1
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603
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In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?
Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object $X\...
- 54.6k
16
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1
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431
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Are there small examples of non-pivotal finite tensor categories?
I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.
Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
- 28.1k
13
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4
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5k
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What is the universal enveloping algebra?
Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
- 12.3k
5
votes
5
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767
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What are the correct axioms for a "weakly associative monoidal functor"?
Definitions and the main question
Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties):
a ...
- 54.6k
7
votes
1
answer
796
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Is there a relative version of Tannakian reconstruction?
According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor ...
- 25.6k
4
votes
1
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341
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When does a certain natural construction on monoidal categories yield a Hopf algebra?
Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of $\...
- 54.6k
15
votes
6
answers
2k
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What structure on a monoidal category would make its 2-category of module categories monoidal and braided?
So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...
- 44.7k
15
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3
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855
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Are supervector spaces the representations of a Hopf algebra?
Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...
- 118k
4
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3
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751
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What is a formula for the "group-like Drinfeld element"?
Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
- 44.7k