All Questions
Tagged with quantum-groups mp.mathematical-physics
20 questions
8
votes
1
answer
313
views
How does the Tannaka duality work for weak Hopf algebras and fusion categories?
I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
- 727
5
votes
1
answer
194
views
In the rep theory of Quantum Double, why does the fusion of 2 "pure fluxes" yield a "pure charge"?
Physicist here, so my notation may be different from standard math notation.
For the quantum double $D(G)$ of a group $G$, we may write representations of $D(G)$ in the following way: Consider a ...
- 569
0
votes
0
answers
105
views
A variant of quantum harmonic oscillators
We have the following variant of harmonic oscillators.
$$
\left\{
\begin{array}{**lr**}
T = a + a^\dagger\\
a | n \rangle = \sqrt{[n]} |n-1 \rangle \\
a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
- 105
5
votes
1
answer
905
views
What is the difference between the Yang--Baxter equation and the quantum Yang--Baxter equation?
For a vector space $V$ and a linear operator $R:V \otimes V \to V \otimes V$, we say that $R$ satisfies the Yang--Baxter equation if
$$(R\otimes id)(id\otimes R)(R\otimes id)
= (id\otimes R)(R\otimes ...
- 1,144
12
votes
2
answers
782
views
Relations between quantum groups at roots of unity, modular representation theory, and physics
I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations ...
- 2,964
3
votes
1
answer
241
views
F-symbols for compact Lie groups
Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, ...
- 195
9
votes
0
answers
353
views
Quantum Hamiltonian reduction and Quantum Airy structure
I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section ...
- 425
5
votes
1
answer
575
views
Is there another quantum deformation of sl(2)?
By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are:
$$
[E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F,
$$
some ...
- 53
32
votes
1
answer
2k
views
Limiting representation theory of quantum groups at roots of unity and $SL(2,\mathbb{C})$
Let $V_N$ denote the $N$-dimensional representation of the quantum group $U_q(\mathfrak s\mathfrak l_2)$. I am told that in the limit $N\to\infty$ with $q=e^{2\pi i/n}$ and $N/n\to\alpha\in(0,1)$, ...
- 18.7k
3
votes
1
answer
145
views
Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?
This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined ...
- 385
12
votes
2
answers
2k
views
Is there any published physics article where $q$-mathematics is applied?
Excuse me for the concern, but I want to ask you a question.
In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science ...
- 551
4
votes
1
answer
508
views
Why does the Bogolyubov transformation work? - In language of Clifford Algebras?
Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb{...
- 605
4
votes
1
answer
566
views
How to obtain a classical r-matrix from a quantum R-matrix?
Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.
- 6,201
1
vote
0
answers
71
views
Low-dimensional classical r-matrices
Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...
- 6,201
2
votes
1
answer
236
views
How to compute $t_0$ and $r^0$ in Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations?
I tried to understand Belavin-Drinfeld's classification of solutions of classical Yang-Baxter equations.
In the book a guide to quantum groups, on page 83, there is an example of solutions of the ...
- 6,201
8
votes
1
answer
432
views
Brauer-Picard for a fusion category coming from a quantum group
In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...
- 993
6
votes
1
answer
308
views
Compact Quantum Groups and FRT-Algebras
As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...
- 159
2
votes
2
answers
327
views
Deformation quantization of a closed Riemann surface with genus >1
Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...
- 309
1
vote
1
answer
184
views
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?
What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...
- 6,201
13
votes
1
answer
848
views
Are Turaev--Viro invariants secretly a discretized path integral?
Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\...
- 18.7k