All Questions
24 questions
2
votes
0
answers
103
views
Morphism of discrete quantum groups
In the paper Kazhdan's Property T for Discrete Quantum Groups
, we read the following fragment:
First, note that I think there is a typo and that codomain and domain of the dual maps have to be ...
- 175
1
vote
1
answer
80
views
Show that if $V\in M(B_0(H)\otimes A)$, then $V(B(H)\otimes 1)V^*\not\subseteq B(H)\otimes A$ where $A$ is a specific unital $C^*$-algebra
Let $\mathbb{G}$ be a compact (quantum group) with function algebra $(C(\mathbb{G}), \Delta)$ and Haar state $\varphi_{\mathbb{G}}$. Consider the associated GNS-representation $\pi_{\mathbb{G}}: C(\...
- 175
3
votes
1
answer
116
views
Unitary in adjointable operators associated with equivariant Hilbert module
Consider the following fragment from the article "Tannaka–Krein duality for compact quantum
homogeneous spaces. I. General theory" by De Commer and Yamashita:
How exactly is $\mathcal{E}\...
- 175
4
votes
1
answer
167
views
Reference request: decomposability of $\mathbb{G}$-Hilbert modules
Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action
$$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \...
- 525
3
votes
1
answer
229
views
Relating different definitions of dual of a compact quantum group
Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $*$-...
- 175
1
vote
0
answers
137
views
Representation of quantum groups
Let $\mathbb{G}=(A,\Delta_A)$ be a C*-quantum group and $\mathbb{H}=(B,\Delta_B)$ be a closed quantum subgroup of $\mathbb{G}$. We say that $\mathbb{H}$ is a closed quantum subgroup of $\mathbb{G}$ if ...
- 121
3
votes
1
answer
163
views
Norm antipode on a Kac-type compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ ...
- 175
3
votes
1
answer
213
views
Woronowicz characters are multiplicative
I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset.
Let $G$ be a $C^*$-algebraic compact quantum group with function algebra $(C(G), \Delta)...
- 175
4
votes
0
answers
165
views
Tensor product of representations on a compact quantum group
Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$.
Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
- 175
5
votes
1
answer
209
views
Subrepresentations of C*-algebraic compact quantum groups
Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-...
- 175
3
votes
1
answer
142
views
Nonstandard Podles spheres as $U_c(\frak{h})$ invariants
In this paper Podles introduced a $2$-parameter family of $q$-deformed spheres $S_{q,c}$ that are now called the "Podles spheres". The case of $c=0$ is very special and is known as the "...
- 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 $*$-...
user167952
5
votes
1
answer
181
views
Matrix coefficients of a compact quantum group
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz).
Definition: A corepresentation matrix of $(A, \Delta)$ is a matrix $a=(a_{i,j}) \in M_n(A)$ such that
$$\...
user167952
9
votes
1
answer
585
views
Finite compact quantum groups
Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we ...
user167952
5
votes
1
answer
202
views
Relating different constructions of the universal compact quantum group
Before asking my question, let me give the necessary background. Readers that are comfortable with the language of universal and reduced compact quantum groups may skip the following two sections.
...
user167952
6
votes
1
answer
338
views
Invertible elements of the Hopf algebra quantum $SU(2)$
Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see
https://en.wikipedia.org/wiki/Compact_quantum_group
(Note that on the ...
- 1,144
2
votes
1
answer
393
views
Peter-Weyl theorem (compact quantum groups)
I'm reading the paper Notes on compact quantum groups. In this paper, the following theorem is proven:
Question: Why is the marked equality true?
user167952
0
votes
1
answer
177
views
Direct sum of representations of a compact quantum group
Let $(A, \Delta)$ be a compact quantum group and $\{(H_\alpha, v_\alpha)\}$ be a collection of representations of $A$. That is,
$$v_\alpha \in M(B_0(H_\alpha) \otimes A); \quad \quad(\text{id}\otimes \...
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
2
votes
2
answers
477
views
Comultiplication of elements of partition of unity
Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra).
Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
- 1,027
12
votes
0
answers
285
views
Is there a non-Kac complex finite dimensional semisimple Hopf algebra?
A complex (finite-dimensional) Hopf algebra is said to be a
Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (...
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 $n \geq 3$?
