Braided Hopf algebras and Quantum Field Theories

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.

I think that -apart from the applications in CFT and TFT already mentioned in previous answers- one of the most fundamental applications of braided Hopf algebras (with both non-trivial and "calculable" braiding), which underlies a significant part of various quantum field theories, is the mathematical foundations of the idea of supersymmetry itself: the notions of super vector space ($\mathbb{Z}_2$-graded vector space) and superalgebra ($\mathbb{Z}_2$-graded algebra) and their "super" tensor products can be conceptually understood in the framework of braided monoidal categories as applications of the (unique) non-trivial braiding of the category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$ of representations of the quasitriangular group Hopf algebra $\mathbb{CZ}_2$.

This can be understood better if you take into account that the following statements:

• $V$ is a $\mathbb{Z}_{2}$-graded vector space (equivalently: a super vector space).

• $V$ is a $\mathbb{CZ}_{2}$-module, through the $\mathbb{Z}_{2}$-action \begin{equation} \begin{array}{cccc} 1 \cdot v = v & & & g \cdot v = (-1)^{|v|}v=\left\{ \begin{array}{r} v, \ \ v\in V_0 \\ -v, \ \ v\in V_1 \end{array} \right. \\ \end{array} \end{equation} for any homogeneous element $v\in V$, where $\ |v| \ $ stands for the degree of $av$. (In other words $|v|=0$, if $v\in V_{0}$ ($v$ is an even element) and $|v|=1$, if $v\in V_{1}$ ($v$ is an odd element)).

• $V$ is a vector space in the braided monoidal Category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$ of representations of the group Hopf algebra $\mathbb{CZ}_{2}$.

are equivalent.

Furthermore, the above correspondence generalizes to superalgebras: The statements:

• $A$ is a $\mathbb{Z}_{2}$-graded algebra (equivalently: a super algebra).

• $A$ is a $\mathbb{CZ}_{2}$-module algebra through the $\mathbb{Z}_{2}$-action \begin{equation} \begin{array}{cccc} 1 \cdot a = a & & & g \cdot a = (-1)^{|a|}a=\left\{ \begin{array}{r} a, \ \ a\in A_0 \\ -a, \ \ a\in A_1 \end{array} \right. \\ \end{array} \end{equation} for any homogeneous element $a\in A$, where $\ |a| \ $ stands for the degree of $a$. (In other words $|a|=0$, if $a\in A_{0}$ ($a$ is an even element) and $|a|=1$, if $a\in A_{1}$ ($a$ is an odd element)).

• $A$ is an algebra in the braided monoidal Category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$ of representations of the group Hopf algebra $\mathbb{CZ}_{2}$

are equivalent.

In the above, the braiding $\Psi$, of the braided monoidal Category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$ (i.e. the Category of $\mathbb{CZ}_{2}$-modules), is given by the family of natural isomorphisms $\psi_{V,W}: V\otimes W \cong W\otimes V$ explicitly written: $$ \psi_{V,W}(v\otimes w)=(-1)^{|v| \cdot |w|} w \otimes v $$ It can furthermore been shown, that the above (non-trivial) braiding is induced by the non-trivial quasitriangular structure of the group Hopf algebra $\mathbb{CZ}_{2}$, given by the $R$-matrix: \begin{equation} R_{\mathbb{Z}_{2}} =\sum R_{\mathbb{Z}_{2}}^{(1)} \otimes R_{\mathbb{Z}_{2}}^{(2)}= \frac{1}{2}(1 \otimes 1 + 1 \otimes g + g \otimes 1 - g \otimes g) \end{equation} To be more specific, this $R$-matrix, induces the -the above mentioned- braiding through $$ \psi_{V,W}(v \otimes w) = \sum (R_{\mathbb{Z}_{2}}^{(2)} \cdot w) \otimes (R_{\mathbb{Z}_{2}}^{(1)} \cdot v)=(-1)^{|v| \cdot |w|} w \otimes v $$ In the above, $v,w$ are any elements of the $\mathbb{CZ}_{2}$-modules (super vector spaces according to the above) $V,W$.

Now, the so-called super tensor product algebra or $\mathbb{Z}_2$-graded tensor product algebra, of superalgebras, is the superalgebra $A\underline{\otimes} B$, whose multiplication $m_{A\underline{\otimes} B}$ given by $$ m_{A\underline{\otimes} B}=(m_{A} \otimes m_{B})(Id \otimes \psi_{B,A} \otimes Id): A \otimes B \otimes A \otimes B \longrightarrow A \otimes B $$ or equivalently: $$ (a \otimes b)(c \otimes d) = (-1)^{|b| \cdot |c|}ac \otimes bd $$ where $A,B$ are superalgebras, $m_A, m_B$ are their multiplications, $b,c$ are homogeneous elements of $B$ and $A$ respectively and $a,d$ any elements of $A$ and $B$ respectively.
(for further details and the generalization of the above for any finite abelian group, you can see this article, sect. $3$, pages 78-81).

Remark: It is interesting to note that most of the formalism of super vector spaces, superalgebras and their super tensor products was known to mathematicians since the late $'40$'s and the idea of supersymmetry in physics dates back to the $'70$'s. However it was not until the advent of quasitriangular Hopf algebras (in the late $'80$'s) and the investigation of their relation to the braided monoidal categories that the above connections were realized. Since, it is usual to speak (mainly in the math. physics literature) rather of "braided" than "graded" tensor products.

Notice also, that the above application involves the simplest non-trivial braiding. This stems from the non-trivial $R$-matrix $R_{\mathbb{Z}_{2}}$ of the group Hopf algebra $\mathbb{CZ}_{2}$ rather than from its trivial quasitriangular structure $R=1\otimes 1$ (i.e. its cocommutativity).

Finally, if you are further interested on the mathematical basis of supersymmetry and its importance in physics, among others I would recommend:

• Supersymmetry for Mathematicians: an Introduction
• these notes

Higher-dimensional algebra I: Braided monoidal 2-categories, John Baez and Martin Neuchl.

Braided monoidal categories are especially interesting because they give efficient procedures for constructing tangle invariants and 3-dimensional topological quantum field theories, and braided monoidal 2-categories have analogous applications to 2-tangle invariants and 4-dimensional TQFT's.