A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations

We give an account of the current state of the approch to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic structures here in one place which are either scattered over the literature, or only implicit in previous writings. In particular we point out mathematical structures which can be helpful to farther develop our mathematical understanding of quantum...

We review the appearance of Hopf algebras in the renormalization of quantum field theories and in the study of diffeomorphisms of the frame bundle important for index computations in noncommutative geometry.

We define a natural concept of duality for the h-Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a deformation both of the function algebra F(SL(2)) and of the enveloping algebra U(sl(2)). Matrix elements of the self-duality in the Peter-Weyl basis are 6j-symbols; this leads to a new algebraic interpretation of the hexagon identity or quantum dyn...

Let $k_q[x, x^{-1}, y]$ be the localization of the quantum plane $k_q[x, y]$ over a field $k$, where $0\neq q\in k$. Then $k_q[x, x^{-1}, y]$ is a graded Hopf algebra, which can be regarded as the non-negative part of the quantum enveloping algebra $U_q({\mathfrak sl}_2)$. Under the assumption that $q$ is not a root of unity, we investigate the coalgebra automorphism group of $k_q[x, x^{-1}, y]$. We describe the structures of the graded coalgebra automorphism group and the co...

We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of po...

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov ($q$-KZ) equation for $U_{\mathsf v}(A_1^{(1)})$ with generic spins. Namely we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the $R$-matrix, or the quantum $6j$ symbols. On the other hand, we prove that the $K$ theoretic Nekrasov partition f...

Let $\mathcal{W}_N$ be a quantized Borel subalgebra of $U_q(sl(2,\mc))$, specialized at a primitive root of unity $\omega = \exp(2i\pi/N)$ of odd order $N >1$. One shows that the $6j$-symbols of cyclic representations of $\mathcal{W}_N$ are representations of the canonical element of a certain extension of the Heisenberg double of $\mathcal{W}_N$. This canonical element is a twisted $q$-dilogarithm. In particular, one gives explicit formulas for these $6j$-symbols, and one co...

A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra. In particular we find a generalized Cartan identity that holds on the whole quantum universal enveloping algebra of the left-invaria...

We construct a universal trigonometric solution of the Gervais-Neveu-Felder equation in the case of finite dimensional simple Lie algebras and finite dimensional contragredient simple Lie superalgebras.

We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ) with values in a tensor product of irreducible sl(2) modules, the equation defined in terms of rational R-matrices. We solve the equation in terms of multidimensional q-hypergeometric integrals. We identify the space of solutions of the qKZ equation with the tensor product of the corresponding modules over the quantum group $U_qsl(2)$. We compute the monodromy of the qKZ equation in terms of the trigon...