Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation

Quantum doubles of finite group algebras form a class of quasi-triangular Hopf algebras which algebraically solve the Yang--Baxter equation. Each representation of the quantum double then gives a matrix solution of the Yang--Baxter equation. Such solutions do not depend on a spectral parameter, and to date there has been little investigation into extending these solutions such that they do depend on a spectral parameter. Here we first explicitly construct the matrix elements of the generators for all irreducible representations of quantum doubles of the dihedral groups $D_n$. These results may be used to determine constant solutions of the Yang--Baxter equation. We then discuss Baxterisation ans\"atze to obtain solutions of the Yang--Baxter equation with spectral parameter and give several examples, including a new 21-vertex model. We also describe this approach in terms of minimal-dimensional representations of the quantum doubles of the alternating group $A_4$ and the symmetric group $S_4$.Comment: 19 pages, no figures, changed introduction, added reference

Solutions of the Yang-Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras

The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev-Zamolodchikov model are discussed.Comment: 34 pages, 2 figure

Bethe ansatz solution of an integrable, non-Abelian anyon chain with D(D_3) symmetry

The exact solution for the energy spectrum of a one-dimensional Hamiltonian with local two-site interactions and periodic boundary conditions is determined. The two-site Hamiltonians commute with the symmetry algebra given by the Drinfeld double D(D_3) of the dihedral group D_3. As such the model describes local interactions between non-Abelian anyons, with fusion rules given by the tensor product decompositions of the irreducible representations of D(D_3). The Bethe ansatz equations which characterise the exact solution are found through the use of functional relations satisfied by a set of mutually commuting transfer matrices.Comment: 19 page

Universal Baxterization for Z\mathbb{Z}-graded Hopf algebras

We present a method for Baxterizing solutions of the constant Yang-Baxter equation associated with $\mathbb{Z}$-graded Hopf algebras. To demonstrate the approach, we provide examples for the Taft algebras and the quantum group $U_q[sl(2)]$.Comment: 8 page

Patient-Reported Outcomes following Single- and Multiple-Radius Total Knee Replacement: A Randomized, Controlled Trial

Although single-radius (SR) designs of total knee replacement (TKR) have theoretical benefits, the clinical advantage conferred by such designs is unknown. The aim of this randomized, controlled study was to compare the short-term clinical outcomes of the two design rationales. A total of 105 knees were randomized to receive either a single radius (Scorpio, Stryker; SR Group) or multiple radius (AGC, Zimmer Biomet; MR group) TKR. Patient-reported outcomes (Oxford Knee Score [OKS] and Knee Society Score [KSS]) were collected at 6 weeks, 6 months, and 1 year following surgery. No knees were revised. There was no difference in primary outcomes: OKS was 39.5 (95% confidence interval [CI]: 36.9–42.1) in the SR group and 38.1 (95% CI: 36.0–40.3) in the MR group (p = 0.40). KSS was 168.4 (95% CI: 159.8–177.0) in the SR group; 159.5 (95% CI 150.5–168.5) in the MR group (p = 0.16). There was a small but statistically significant difference in the degree of change of the objective subscale of the KSS, favoring the SR design (p = 0.04), but this is of uncertain clinical relevance. The reported benefits of SR designs do not provide demonstrable functional advantages in the short term

Framework for classifying logical operators in stabilizer codes

Entanglement, as studied in quantum information science, and non-local quantum correlations, as studied in condensed matter physics, are fundamentally akin to each other. However, their relationship is often hard to quantify due to the lack of a general approach to study both on the same footing. In particular, while entanglement and non-local correlations are properties of states, both arise from symmetries of global operators that commute with the system Hamiltonian. Here, we introduce a framework for completely classifying the local and non-local properties of all such global operators, given the Hamiltonian and a bi-partitioning of the system. This framework is limited to descriptions based on stabilizer quantum codes, but may be generalized. We illustrate the use of this framework to study entanglement and non-local correlations by analyzing global symmetries in topological order, distribution of entanglement and entanglement entropy.Comment: 20 pages, 9 figure