All Questions
9 questions
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
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
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
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
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
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