Z2×Z2 \mathbb{Z}_2 \times \mathbb{Z}_2 generalizations of N=1{\cal N} = 1 superconformal Galilei algebras and their representations

We introduce two classes of novel color superalgebras of $\mathbb{Z}_2 \times \mathbb{Z}_2$ grading. This is done by realizing members of each in the universal enveloping algebra of the ${\cal N}=1$ supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded setting.Comment: 17 pages, no figur

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

Modeling the dynamics of a tracer particle in an elastic active gel

The internal dynamics of active gels, both in artificial (in-vitro) model systems and inside the cytoskeleton of living cells, has been extensively studied by experiments of recent years. These dynamics are probed using tracer particles embedded in the network of biopolymers together with molecular motors, and distinct non-thermal behavior is observed. We present a theoretical model of the dynamics of a trapped active particle, which allows us to quantify the deviations from equilibrium behavior, using both analytic and numerical calculations. We map the different regimes of dynamics in this system, and highlight the different manifestations of activity: breakdown of the virial theorem and equipartition, different elasticity-dependent "effective temperatures" and distinct non-Gaussian distributions. Our results shed light on puzzling observations in active gel experiments, and provide physical interpretation of existing observations, as well as predictions for future studies.Comment: 11 pages, 6 figure

The influence of clearance on friction, lubrication and squeaking in large diameter metal-on-metal hip replacements

Large diameter metal-on-metal bearings (MOM) are becoming increasingly popular, addressing the needs of young and more active patients. Clinical data has shown excellent short-to-mid-term results, though incidences of transient squeaking have been noted between implantation and up to 2 years post-operative. Geometric design features, such as clearance, have been significant in influencing the performance of the bearings. Sets of MOM bearings with different clearances were investigated in this study using a hip friction simulator to examine the influence of clearance on friction, lubrication and squeaking. The friction factor was found to be highest in the largest clearance bearings under all test conditions. The incidence of squeaking was also highest in the large clearance bearings, with all bearings in this group squeaking throughout the study. A very low incidence of squeaking was observed in the other two clearance groups. The measured lubricating film was found to be lowest in the large clearance bearings. This study suggests that increasing the bearing clearance results in reduced lubricant film thickness, increased friction and an increased incidence of squeaking

Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2 generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions

We introduce a class of novel $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded color superalgebras is presented. It turns out that infinitely many members of the class have non-trivial extensions. We also demonstrate that the color superalgebras (with and without central extensions) have adjoint and superadjoint operations.Comment: 19 pages, no figure, Revision in Section 2 and 3. Some new reference