Braided Hopf Algebras and Quantum Field Theories – Mathematical Physics

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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".

Best Answer

Some particular braided Hopf algebras known as Nichols algebras 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

  • 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

  • Semikhatov, A. M.; Tipunin, I. Yu. The Nichols algebra of screenings. Commun. Contemp. Math. 14 (2012), no. 4, 1250029, 66 pp. MR2965674, arXiv

Added:

  • Lentner, S. Quantum groups and Nichols algebras acting on conformal field theories, arXiv

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.

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