All Questions
21 questions
3
votes
0
answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
3
votes
0
answers
74
views
Are all enveloping algebras $\mathcal{U}(\mathfrak{g})$ locally compact quantum groups?
Let us consider the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$.
Under what assumptions about $\mathfrak{g}$, does the enveloping algebra generate a locally ...
- 143
1
vote
0
answers
70
views
Affiliating the whole algebra of 'coordinates' with a locally compact quantum group
When constructing a quantum deformation of a classical (matrix) locally compact group, we usually start with the *-Hopf algebra $A$ of matrix entries/coordinates. We then deform this algebra ($A_q$) ...
- 143
6
votes
2
answers
198
views
Proof that every commutative locally compact quantum group arises from a locally compact group
It is well-known that there is a bijection (up to isomorphisms) between locally compact quantum groups whose algebra is commutative and classical locally compact groups. I seem to cannot find a proof ...
- 143
5
votes
1
answer
239
views
Completely isometric coaction of discrete quantum group is multiplicative?
Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense ...
- 525
6
votes
1
answer
172
views
Norm of contragredient of unitary representations of compact quantum groups
Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a ...
- 960
7
votes
1
answer
345
views
Is the comultiplication of a compact quantum group always injective?
Let $(A, \Delta)$ be a compact quantum group in the sense of Woronowicz. Is it true that the comultiplication $\Delta : A \to A \otimes A$ always injective?
This is true for both the universal (...
- 1,586
9
votes
1
answer
585
views
Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
user167952
5
votes
1
answer
202
views
Relating different constructions of the universal compact quantum group
Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...
user167952
0
votes
1
answer
261
views
Definition intertwiner of representations of compact quantum groups
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
user167952
2
votes
2
answers
217
views
Kernel of intertwiner is invariant (compact quantum groups)
Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $...
user167952
0
votes
1
answer
158
views
Showing a product on a character space is continuous
Quoting from Timmermann's An invitation to quantum groups and duality:
Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact
quantum group. Then there exists a compact group $G$ and ...
- 1,027
5
votes
1
answer
228
views
Zero divisors in compact quantum groups
Let $\mathcal{G}$ be compact quantum group in the sense of S. L. Woronowicz. As is well-known, every compact quantum group contains a dense Hopf algebra, called the polynomial Hopf algebra Pol$(\...
- 969
9
votes
1
answer
207
views
Separability of compact quantum groups
In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
- 151
3
votes
1
answer
187
views
Number of Isomorphism Classes of Corepresentations of A Compact Quantum Group
Given a compact quantum group $(G,\Delta)$, with dense Hopf algebra $H$, is it always true that, up to isomorphism, $H$ will have a countable number of irreducible comodules?
- 159
5
votes
1
answer
283
views
Reference request quantum SU(3)
Woronowicz shows that the C*-algebras of quantum $SU(2)$ are isomorphic (only as C*-algebras, forgetting the quantum group structure). Are there similar results for quantum $SU(n)$ for $n \geq 3$?
5
votes
1
answer
203
views
What are the applications of the depth 2 reduction to the subfactors theory?
Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth ...
1
vote
0
answers
228
views
Clebsch Gordan coefficients of compact quantum groups
Consider a compact quantum group $G$. Let $a, b$ and $c$ be irreducible unitary corepresentations and assume that $c$ is contained in $a \otimes b$. Let $U$ be the intertwiner from the representation ...
- 11
3
votes
0
answers
229
views
The convolution on a semisimple finite quantum groupoid
Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{...
3
votes
1
answer
248
views
example of a compact quantum group at a root of unity?
In Woronowicz's theory of compact quantum groups, the most well-known example is $SU_q(2)$, for $q$ a real number. Moreover, all the other examples of compact quantum groups, based some Drinfeld--...
- 31
7
votes
4
answers
1k
views
Compact Quantum Groups from Hopf Algebras
For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around,...
- 1,462