All Questions
Tagged with hopf-algebras ct.category-theory
66 questions
5
votes
0
answers
92
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$\text{Rep}(D_4)$ and its three fiber functors
It is well-known that the fusion category $\text{Rep}(D_4)$ of representations of the dihedral group $D_4$ of order 8 admits three distinct fiber functors. Therefore, there are three different Hopf ...
3
votes
0
answers
87
views
Functorial relationships between Hopf algebras and rough paths
In rough paths theory, the signature of a path $x: [0, T] \to \mathbb{R}^d$ is an element in the tensor algebra $T((\mathbb{R}^d))$. These signatures reside within the group-like elements and ...
- 232
5
votes
1
answer
179
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
0
votes
0
answers
44
views
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 ...
4
votes
0
answers
68
views
Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
- 727
5
votes
0
answers
130
views
The Balmer spectrum and the thick tensor ideals of the derived category of a Hopf algebra
Given a Hopf algebra $H$ over a field $\mathbb{k}$, the category of finite-dimensional left-$H$-modules naturally becomes a rigid monoidal category with exact monoidal product. Thus clearly the ...
- 1,474
6
votes
0
answers
122
views
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
0
votes
1
answer
294
views
Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
user525442
3
votes
2
answers
135
views
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
votes
0
answers
98
views
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
6
votes
1
answer
207
views
Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
- 764
3
votes
0
answers
133
views
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
9
votes
0
answers
326
views
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
1
vote
0
answers
121
views
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 $\...
- 113
3
votes
0
answers
113
views
Is the Frobenius property invariant by Morita equivalence?
Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....
3
votes
0
answers
276
views
Is there a non-pointed simple integral modular fusion category?
The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds:
Statement 1 (open): There is a ...
7
votes
1
answer
315
views
Does the category of commutative and cocommutative Hopf algebras have enough injectives?
It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
- 3,079
8
votes
2
answers
852
views
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 ...
5
votes
1
answer
473
views
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
views
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
14
votes
3
answers
1k
views
What is known about the category of Hopf algebras?
Several weeks ago I asked this at MathStackExchange, and to my surprise nobody answered.
Recently I understood that I know almost nothing about the category $\operatorname{HopfAlg}$ of Hopf algebras (...
- 7,426
5
votes
0
answers
257
views
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
0
votes
0
answers
78
views
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
3
votes
0
answers
119
views
Is the category of Yetter-Drinfeld modules abelian?
Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
- 1,144
4
votes
0
answers
208
views
Are the finite quantum permutation groups, weakly group-theoretical?
Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
7
votes
2
answers
631
views
Abelian category from the category of Hopf algebras
The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf
sub-algebra of $H_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same ...
- 1,144
4
votes
0
answers
106
views
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 ...
- 1,813
3
votes
1
answer
456
views
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
5
votes
2
answers
462
views
Subfunctor of internal Hom
Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...
- 353
6
votes
0
answers
338
views
Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?
Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
- 63.9k
10
votes
3
answers
856
views
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
views
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
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 ...
- 835
3
votes
0
answers
70
views
Is there a semisimple Hopf algebra Grothendieck equivalent to a strictly weak one?
By Corollary 2.22 in On fusion categories (by Pavel Etingof, Dmitri Nikshych and Viktor Ostrik) any fusion category is equivalent to the category of finite dimensional representations of a semisimple ...
26
votes
1
answer
2k
views
Have the Quantum Group Theorists taught the Group Theorists Anything?
I will start with the general before moving to the specific.
Consider for a moment the two (very) soft definitions.
An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
- 1,027
4
votes
0
answers
159
views
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
9
votes
4
answers
1k
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.
- 993
9
votes
1
answer
414
views
What additional property does the antipode give on the category of all modules over an Hopf algebra?
It is well known that many constructions involving bialgebras extends to monoidal categories, and often becomes more natural in that framework.
If one cares about the category of finite dimensional ...
- 8,524
1
vote
0
answers
138
views
When is a Frobenius Algebra a Quasi-Frobenius Ring?
Let $F$ be Frobenius algebra in the monoidal category $\mathcal{C}$ of bimodules over a not-necessarily commutative algebra $A$. When is it true that $F$ is a quasi-Frobenius ring.
For example, this ...
- 459
0
votes
0
answers
92
views
$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]
Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space,
but the statement is easier if we also suppose $X$ ...
- 3,051
4
votes
0
answers
130
views
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
8
votes
0
answers
217
views
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
4
votes
1
answer
260
views
Do the modules over a Hopf algebra in a braided monoidal category form a monoidal category?
Given a (strict) braided monoidal category $(\mathcal{C},\otimes,I)$ with braiding $b$ and a Hopf algebra $H$ in $\mathcal{C}$. There is a category Rep($H$) of modules over $H$ in $\mathcal{C}$. Do ...
- 449
4
votes
0
answers
310
views
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
5
votes
1
answer
347
views
Category of bicomodules of a cosemisimple Hopf algebra
A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
- 459
3
votes
1
answer
181
views
An interpretation of this construction giving an operad from a bialgebra?
Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = ...
- 5,040
4
votes
1
answer
347
views
Fusion Rules for Quantum Groups
For the Drinfeld--Jimbo quantum groups $U_q(\frak{g})$, we have an equivalence of categories between the representations of $U_q(\frak{g})$ and the representations of $U(\frak{g})$.
Is this a ...
- 1,479
9
votes
2
answers
1k
views
Algebra in a category
I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \...
- 6,201
2
votes
1
answer
249
views
Yetter-Drinfeld modules as rigid category
I'm reading a proof of the following theorem
If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
In this proof we define $V^*...
- 797
21
votes
2
answers
3k
views
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