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3 votes
0 answers
65 views

A combinatorial Dyson-Schwinger equation, tree diagrams, and compositional inversion of a Laurent series

In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9: Combinatorial Dyson-Schwinger equations are functional equations with solutions in $...
5 votes
1 answer
80 views

Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricted subspaces of $V^{\otimes N}$

An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ where ...
8 votes
1 answer
312 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 ...
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 ...
17 votes
4 answers
1k views

Braided Hopf algebras and Quantum Field Theories

It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...
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 ...
13 votes
2 answers
1k views

Hopf Algebra for a physicist

Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
19 votes
1 answer
2k views

Connes-Kreimer Hopf algebra and cosmic Galois group

Hi, I'm interested in the relation between the two following constructions motivated by renormalization: Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...