All Questions
Tagged with oa.operator-algebras qa.quantum-algebra
67 questions
22
votes
2
answers
2k
views
Non weakly-group-theoretical integral fusion category
Is there an integral fusion category of rank $7$, FPdim $210$ and type $(1,5,5,5,6,7,7)$ with the following fusion rules (or the little $\color{purple}{\text{variation}}$ below)?
$$\scriptsize{\begin{...
17
votes
1
answer
2k
views
The cyclic subfactors theory: a quantum arithmetic?
Context: First recall some results:
Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-Popa ...
14
votes
1
answer
880
views
Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories
Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all $...
- 195
13
votes
1
answer
1k
views
Why is Planar algebras I (by Vaughan Jones) not published?
On Saturday 4 September 1999, Vaughan Jones put on arXiv a paper entitled Planar algebras, I.
Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the ...
13
votes
3
answers
686
views
Does every Frobenius algebra in a monoidal *-category give a Q-system?
Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...
- 28.1k
9
votes
2
answers
356
views
Is there a subfactor construction involving 2-groups?
I seem to recall that there is a straightforward subfactor construction that yields fusion categories given by G-graded vector spaces and representations of G, for finite groups G. Is there an ...
- 45.7k
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
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
9
votes
3
answers
409
views
What are the intermediate subfactors of the tensor product of two maximal subfactors?
Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...
9
votes
0
answers
268
views
Existence/characterization/properties of $C^*$-algebras which "are" quantization of compact symplectic manifolds?
Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...
- 24.9k
8
votes
1
answer
482
views
Eigenvalues of the free sphere
Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x_1,\ldots,x_n:\mathbb R^n\to\mathbb R$, and in fact we have the ...
- 1,363
8
votes
0
answers
488
views
Is there a non-trivial Hopf algebra without left coideal subalgebra?
Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\...
8
votes
0
answers
306
views
Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?
A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Question: Are there only finitely many maximal irreducible amenable subfactors at ...
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
7
votes
4
answers
1k
views
Quantum channels as categories: question 1.
A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...
- 283
7
votes
2
answers
409
views
Are subfactor planar algebras hard to classify at index 6?
Given a finite index inclusion, $N\subset M$, of $II_1$ factors we can construct two towers of finite dimensional algebras known as the $\textit{standard invariant}$. For low index, this has allowed ...
- 2,441
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
7
votes
3
answers
698
views
Jordan-Hölder theorem for subfactors?
All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.
First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
7
votes
0
answers
169
views
How to translate connection on four graphs to quantum 6j symbols
I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...
- 437
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
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
6
votes
1
answer
321
views
Kernel of the natural map between group $C^*$-algebras
Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...
- 9,330
6
votes
0
answers
128
views
Unitary fusion category and subfactor
From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor.
By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
6
votes
0
answers
239
views
Existence of a Kac algebra for a given fusion ring in a particular class
A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r}...
6
votes
0
answers
369
views
Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
- 16.2k
5
votes
1
answer
152
views
when is an algebra map conjugate to a star algebra map
Take a (unital) algebra map $f:A\to B$ between two unital C* algebras - not necessarily star preserving. Under what circumstances is there a $b\in B$ so that $g(a)=b\ f(a)\ b^{-1}$ is a star algebra ...
- 1,143
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
5
votes
1
answer
177
views
Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?
A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.
Is there an infinite depth irreducible finite index maximal subfactor (other than ...
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
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
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 ...
5
votes
1
answer
261
views
Jordan-Hölder theorem for planar algebras?
First recall the Jordan-Hölder theorem for groups:
Theorem (Jordan-Hölder): Let $G$ be a group, and let $$ G=G_1 \supset G_2 \supset \dots \supset G_r = \{ e \} $$ be a normal tower such that ...
5
votes
1
answer
694
views
Log structure and degeneration
I am interested in compactification of the moduli space of elliptic curves, and I heard that Log geometry is very important for the problem.
I am developping the same technique for quantum geometry.
...
- 53
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$?
4
votes
1
answer
275
views
What are the norms of the generators of the standard Podleś sphere?
Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations
\begin{equation*}
\begin{split}
&a=a^*,~ ...
- 9,009
4
votes
1
answer
272
views
K-Theory of Algebra of Zeroth Order Pseudo differential operators
Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?
Thanx!
- 309
4
votes
2
answers
542
views
The category of subfactors extending the category of groups?
This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
4
votes
0
answers
190
views
A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras
The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:
maximal,
it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
4
votes
0
answers
338
views
Quantization of $S^2$ as $C^*$-algebra?
The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...
- 24.9k
4
votes
0
answers
151
views
Is there a maximal finite depth infinite index irreducible subfactor?
A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...
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
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
3
votes
2
answers
1k
views
Spectral decomposition for an arbitrary linear combination of position and momentum operators
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i $\frac{...
- 195
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
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 ...
- 59
3
votes
0
answers
138
views
Is there a finite depth irreducible subfactor of prime index and not group-subgroup?
Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
3
votes
0
answers
134
views
What are all the possible indices for the finite depth subfactors?
Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
3
votes
0
answers
111
views
Is there a non-irreducible maximal subfactor other than two-sided TLJ?
A subfactor $N \subseteq M$ is called:
irreducible if $N' \cap M = \mathbb{C}$,
maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$.
The two-sided ...
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{...