All Questions
Tagged with qa.quantum-algebra hopf-algebras
180 questions
7
votes
2
answers
201
views
Does the canonical element associated to a finite dimensional $\mathbb{C}^* $-Hopf algebra always have finite order?
Let $\mathcal{A}$ be a finite dimensional $\mathbb{C}^* $-Hopf algebra. Let $B(\mathcal{A})$ be a basis of $\mathcal{A}$ and let
$B(\mathcal{A}^* )=\{ \delta_x\in\mathcal{A}^* | x\in B(\mathcal{A}) \}$...
- 156k
13
votes
1
answer
284
views
Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
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
2
votes
1
answer
78
views
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(...
CommunityBot
- 1
6
votes
0
answers
349
views
Quantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum ...
4
votes
0
answers
82
views
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
$$...
- 727
4
votes
0
answers
68
views
Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
- 727
7
votes
3
answers
650
views
Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
- 51
3
votes
0
answers
276
views
Is there a non-pointed simple integral modular fusion category?
The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds:
Statement 1 (open): There is a ...
3
votes
1
answer
167
views
quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras
In a nutshell, my question is:
Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?
I will now make it more precise. One could define a ...
- 213
4
votes
1
answer
161
views
Existence of an element in a Hopf algebra that satisfies a 'flip' property
Consider a Hopf algebra $H$ where $S$ is the antipode and $\Delta$ is the coproduct. Given $a \in H$, I want to know if there always exists an element $b \in H$ satisfying the 'flip' property: $$\...
- 12.9k
7
votes
0
answers
331
views
An alternative Cauchy theorem on Hopf algebras
Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].
We are interesting in an alternative ...
4
votes
0
answers
56
views
When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?
Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...
- 143
7
votes
0
answers
151
views
How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?
Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
- 193
2
votes
0
answers
28
views
Examples of (weak-)bialgebras/Hopf algebras with a finite dimensional unitary representation and corepresentation and polynomial growth rate
I need examples (the more the better, even better if there is a systematic way of construction) of (weak-)bialgebras or (weak-)Hopf algebras $H$ with a finite dimensional representation $\rho$ and a ...
- 727
3
votes
1
answer
315
views
Is there a definition of Heisenberg double for quasi-Hopf algebras?
$\newcommand{\dmod}{\text{-}\mathrm{mod}}$Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $k$ and $A\dmod$ be the category of finite dimensional left $A$-modules.
Since $A\dmod$ is a ...
- 255
4
votes
1
answer
175
views
Drinfeld-Jimbo quantum groups for $q=0$
In the Wikipedia page of Drinfeld--Jimbo quantum groups the values of $q=0,1$ are excluded so as to avoid dividing by zero. The $q=1$ case is discussed in this old question. What about the $q=0$ case? ...
- 311
11
votes
2
answers
1k
views
Quantized Enveloping Algebras at $q=1$
As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has ...
- 1,144
4
votes
1
answer
228
views
Hopf algebra and coideal question
Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-...
14
votes
2
answers
1k
views
Is a bialgebra with all group-like elements invertible a Hopf algebra?
We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question :
Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g \...
- 2,821
20
votes
10
answers
4k
views
Hopf algebras examples
Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are well-...
- 7,426
3
votes
0
answers
151
views
Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
- 3,446
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....
7
votes
0
answers
385
views
How to define $U_q \mathfrak{g}$ without generators and relations?
I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
- 700
1
vote
1
answer
129
views
About extensions between morphisms on the multiplier algebra
Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism
$$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
CommunityBot
- 1
5
votes
0
answers
203
views
Is there a perfect integral fusion category with PFdim = 2 mod 4?
A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff every $1$-dimensional complex representation of $G$ is trivial.
proof: First if $...
2
votes
1
answer
180
views
Timmerman's "An invitation to quantum groups and duality" corollary 3.2.8
Consider the following propositions of Timmerman's book "An invitation to quantum groups and duality":
I do not understand the equivalence
$$h(\mathfrak{C}(\delta_V)S(\mathfrak{C}(\delta_W))...
- 211
13
votes
6
answers
2k
views
Hopf algebras arising as Group Algebras
Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
- 4,713
16
votes
1
answer
431
views
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 ...
38
votes
6
answers
4k
views
Why Drinfel'd-Jimbo-type quantum groups?
Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
- 63.9k
10
votes
1
answer
570
views
Hopf algebra with a non-invertible antipode
What is an example of a Hopf algebra with a non-invertible antipode?
- 4,600
3
votes
0
answers
103
views
How to interpret compositional diagrams for quantum sets algebraically
$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ...
- 63.9k
3
votes
1
answer
207
views
Characterising algebraic compact quantum groups among Hopf $^*$-algebras
Let $(A, \Delta)$ be a Hopf $^*$-algebra. Assume that $\{u^\alpha\}_{\alpha \in I}$ is a maximal collection of pairwise inequivalent irreducible unitary corepresentation matrices. I want to show that
$...
- 1,986
1
vote
1
answer
97
views
Orthogonality relations for Haar state and antipode (Timmerman)
Consider the following proposition from Timmerman's "An invitation to quantum groups and duality":
I am having trouble seeing why the boxed equations are true (Note that on the left the ...
- 1,037
1
vote
0
answers
62
views
Indecomposable comodules
For a Hopf algebra $A$, we say that a comodule $V$ is indecomposable if it is not equivalent to a direct sum of irreducible comodules.
$\bullet$ What is an example of a finite dimensional ...
- 12.9k
1
vote
1
answer
652
views
What is a coalgebra?
A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
- 28.1k
5
votes
2
answers
343
views
Classifying Hopf algebras that admit a single irreducible comodule
Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k ...
- 7,746
15
votes
1
answer
657
views
Is every finite quantum group a quantum symmetry group?
This post is basically a quantum extension of Is every finite group a group of “symmetries”?
Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra.
Frucht's theorem ...
- 1,037
4
votes
3
answers
540
views
Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion.
Now every Hopf algebra $H$ admits a one-dimensional ...
- 7,746
8
votes
2
answers
853
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 ...
- 63.1k
2
votes
1
answer
121
views
Definition of multiplier bialgebra
Consider the following fragments from "An invitation to quantum groups and duality" by Timmerman:
Question: In remark 2.1.6 (ii), it is stated that the homomorphism $\Delta\otimes \text{id}:...
- 2,385
1
vote
1
answer
157
views
Non-degeneracy of comultiplication (multiplier Hopf algebras)
Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele.
Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \...
- 18.7k
9
votes
1
answer
332
views
The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$
I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
- 7,746
7
votes
2
answers
469
views
Low dimensional noncommutative non-cocommutative Hopf algebras
Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
- 7,746
2
votes
0
answers
99
views
Superfluous axioms for ribbon Hopf algebra
In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying
(1) $v^2 ...
- 601
18
votes
2
answers
1k
views
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 $(A, \Delta, \varepsilon, \Phi)$ ...
- 8,524
0
votes
1
answer
115
views
Antipode on a multiplier Hopf-algebra
Probably an easy question, but here goes:
I'm reading the paper Multiplier Hopf algebras by Van Daele.
Let $(A, \Delta)$ be a multiplier Hopf algebra. Let $L(A), R(A), M(A)$ be the left, right and ...
- 18.7k
2
votes
1
answer
205
views
A comodule algebra map from a Hopf algebra to itself
Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
- 1,144
6
votes
1
answer
194
views
Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
- 4,351
4
votes
1
answer
101
views
Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
- 12.3k