Questions tagged [hopf-algebra]
13 questions
23
votes
7 answers
What are the uses of Hopf algebras in physics?
Hopf algebra is nice object full of structure (a bialgebra with an antipode). To get some idea what it looks like, group itself is a Hopf algebra, considered over a field with one element ;) usual multiplication, diagonal comultiplication, obvious…
Marek
- 23,981
21
votes
2 answers
What is precisely a Yangian symmetry?
The terms Yangian and Yangian symmetry appear in a list of physical problems (spin chains, Hubbard model, ABJM theory, $\mathcal{N}= 4$ super Yang-Mills in $d=4$, $\mathcal{N}= 8$ SUGRA in $d=4$), seem to be linked to (super)conformal symmetries and…
Trimok
- 18,043
6
votes
1 answer
Hopf Algebras in Quantum Groups
In the theory of quantum groups Hopf algebras arise via the Fourier transform:
A third point of view is that Hopf algebras are the next simplest
category after Abelian groups admitting Fourier transform.
At least for nice functions, a Fourier…
bolbteppa
- 4,168
4
votes
0 answers
Hopf algebras vs Fusion categories for topological order
Disclaimer: Before I begin with the question I want to warn that some people would argue that it is a math question and not a physics question. However, it finds it origins in the study of topological order using tensor networks and hence I would…
NDewolf
- 1,415
3
votes
1 answer
Boson calculus and Maximum Weight State
I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with.
Let $\left|0\right\rangle$ denote the Fock vacuum state so that $b_j \left|0\right\rangle = 0$, for all $j$. For any…
Jack
- 205
3
votes
1 answer
Quantum Double Model and Chern-Simons with finite gauge group
Is there a relationship between Kitaev's quantum double model for a finite group $ G $ and a Chern Simons theory with finite gauge group $ G $. They are apparently both related to quantum groups and hopf algebras associated with $ G…
3
votes
1 answer
Relationship between classical $q$-deformed General Relativity and the cosmological constant
The frame-connection formulation of pure General Relativity in 4 dimensions is given by the action
$$
S_{4d}[e, \omega] = \frac{1}{2 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge F^{KL},
$$
where $e^I = e^I_{\mu} dx^{\mu}$ is the frame…
Prof. Legolasov
- 16,527
2
votes
0 answers
Hopfion with only in-plane vortex rather than skyrmion along the torus?
A simple torus-like hopfion with hopf charge $Q_H=1$ will typically exhibit a skyrmion at each slice cutting the toroidal circle. What if the skyrmion is replaced by an in-plane 2D vortex, i.e., we set the vector component $S_3=0$ for all such slice…
xiaohuamao
- 3,971
2
votes
0 answers
External structure in BPHZ/Hopf algebraic renormalization of QED
I'm currently trying to understand and reproduce the Hopf-algebraic renormalization of QED presented by Walter D. Van Suijlekom. I don't understand why he chooses these external structures in (2), (7), (8), (10):
$$
\langle\sigma_{(1)},f\rangle =…
Txordi
- 33
- 4
1
vote
0 answers
To what extent are Hopf algebras relevant in physically understanding quantum field theory?
From what I understand, Hopf algebras are useful algebraic objects for introducing a particular kind of renormalisation.
However, I'm struggling to understand whether this is just a neat high-level algebraic 'trick', or whether it actually has any…
1
vote
0 answers
Are there concrete examples of, and ways to visualise, quantum groups?
I'm trying to get my head around quantum groups, and the basic ideas that (a) they're actually algebras, and (b) they have ''deformed'' relations such as $$ab=ba\rightarrow ab=qba$$ are all fine and good. But I'm struggling to see the point of them.…
StarWombat
- 41
1
vote
2 answers
Why do string theory and Hopf algebra renormalization seem to have no intersection?
Hopf algebra appears in recent papers that systematize renormalization of quantum field theory (QFT). For example see Connes' work and citing papers or a paper referenced here on PSE:
R. E. Borcherds, "Renormalization and quantum field…
Carl Brannen
- 13,059
1
vote
0 answers
When do the solutions of combinatorial Dyson-Schwinger equations generate a Hopf subalgebra?
Say I have a set of combinatorial Dyson-Schwinger equations of the form
$$\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha B_+^n (f_1(X_1,...X_N)) \end{align}$$
that has an unique…
Pxx
- 1,773