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

Structure and phase boundaries of compressed liquid hydrogen

We have mapped the molecular-atomic transition in liquid hydrogen using first principles molecular dynamics. We predict that a molecular phase with short-range orientational order exists at pressures above 100 GPa. The presence of this ordering and the structure emerging near the dissociation transition provide an explanation for the sharpness of the molecular-atomic crossover and the concurrent pressure drop at high pressures. Our findings have non-trivial implications for simulations of hydrogen; previous equation of state data for the molecular liquid may require revision. Arguments for the possibility of a $1^{st}$ order liquid-liquid transition are discussed

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

Statistical analysis of the owl:sameAs network for aligning concepts in the linking open data cloud

The massively distributed publication of linked data has brought to the attention of scientific community the limitations of classic methods for achieving data integration and the opportunities of pushing the boundaries of the field by experimenting this collective enterprise that is the linking open data cloud. While reusing existing ontologies is the choice of preference, the exploitation of ontology alignments still is a required step for easing the burden of integrating heterogeneous data sets. Alignments, even between the most used vocabularies, is still poorly supported in systems nowadays whereas links between instances are the most widely used means for bridging the gap between different data sets. We provide in this paper an account of our statistical and qualitative analysis of the network of instance level equivalences in the Linking Open Data Cloud (i.e. the sameAs network) in order to automatically compute alignments at the conceptual level. Moreover, we explore the effect of ontological information when adopting classical Jaccard methods to the ontology alignment task. Automating such task will allow in fact to achieve a clearer conceptual description of the data at the cloud level, while improving the level of integration between datasets. <br/

Efficient Discrete Approximations of Quantum Gates

Quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision epsilon is only polynomial in log 1/epsilon. Here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/epsilon, a result which matches the lower bound from counting volume up to constant factor.Comment: 7 pages, no figures, v3 revised to correct major error in previous version

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