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 algebras, Hochschild cohomology, Cartier-Quillen cohomology, motivic Galois theory etc. 


But all these structures are based on Hopf algebras in monoidal categories with trivial braiding, e.g. given by $\tau(a\otimes b)=b\otimes a$.


Are there some applications of braided Hopf algebras (i.e. an object in some braided monoidal category) in QFT ? I'm looking for some examples which are both nontrivial and "calculable". 



Braided Hopf algebras and Quantum Field Theories
Some particular braided Hopf algebras known as [Nichols algebras](https://en.wikipedia.org/wiki/Nichols_algebra) are useful in conformal field theories. Here you have some references:


* Semikhatov, A. M.; Tipunin, I. Yu. Logarithmic $\widehat{s\ell}(2)$ CFT models from Nichols algebras: I. J. Phys. A 46 (2013), no. 49, 494011, 53 pp. MR3146017, [arXiv](http://arxiv.org/abs/1301.2235)
* Semikhatov, A. M. Fusion in the entwined category of Yetter-Drinfeld modules of a rank-1 Nichols algebra. Russian version appears in Teoret. Mat. Fiz. 173 (2012), no. 1, 3–37. Theoret. and Math. Phys. 173 (2012), no. 1, 1329--1358. MR3171534, [arXiv](http://arxiv.org/abs/1109.5919)
* Semikhatov, A. M.; Tipunin, I. Yu. The Nichols algebra of screenings. Commun. Contemp. Math. 14 (2012), no. 4, 1250029, 66 pp. MR2965674, [arXiv](http://arxiv.org/abs/1101.5810)


**Added**:


* Lentner, S. Quantum groups and Nichols algebras acting on conformal field theories, [arXiv](https://arxiv.org/abs/1702.06431)


The abstract is the following: 



> 
> We prove a long-standing conjecture by B. Feigin et al. that certain
>  screening operators on a conformal field theory obey the algebra
>  relations of the Borel part of a quantum group (and more generally a
>  diagonal Nichols algebra). Up to now this has been proven only for the
>  quantum group $u\_q(\mathfrak{sl}\_2)$. The proof is based on a novel,
>  intimate relation between Hopf algebras, Vertex algebras and a class
>  of analytic functions in several variables, which are generalizations
>  of Selberg integrals. These special functions have zeroes wherever the
>  associated diagonal Nichols algebra has a relation, because we can
>  prove analytically a quantum symmetrizer formula for them. Morevover,
>  we can use the poles of these functions to construct a crucial Weyl
>  group action. Our result produces an infinite-dimensional graded
>  representation of any quantum group or Nichols algebra. We discuss
>  applications of this representation to Kazhdan-Lusztig theory.
> 
> 
> 


