389 research outputs found

### 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 to the Yang-Baxter Equation and Casimir Invariants for the Quantised Orthosymplectic Superalgebra

For the last fifteen years quantum superalgebras have been used to model
supersymmetric quantum systems. A class of quasi-triangular Hopf superalgebras,
they each contain a universal $R$-matrix, which automatically satisfies the
Yang--Baxter equation. Applying the vector representation to the left-hand side
of a universal $R$-matrix gives a Lax operator. These are of significant
interest in mathematical physics as they provide solutions to the Yang--Baxter
equation in an arbitrary representation, which give rise to integrable models.
In this thesis a Lax operator is constructed for the quantised
orthosymplectic superalgebra $U_q[osp(m|n)]$ for all $m > 2, n \geq 2$ where
$n$ is even. This can then be used to find a solution to the Yang--Baxter
equation in an arbitrary representation of $U_q[osp(m|n)]$, with the example of
the vector representation given in detail.
In studying the integrable models arising from solutions to the Yang--Baxter
equation, it is desirable to understand the representation theory of the
superalgebra. Finding the Casimir invariants of the system and exploring their
behaviour helps in this understanding. In this thesis the Lax operator is used
to construct an infinite family of Casimir invariants of $U_q[osp(m|n)]$ and to
calculate their eigenvalues in an arbitrary irreducible representation.Comment: Approx. 120 pages, no figures, PhD thesi

### SU(3) monopoles and their fields

Some aspects of the fields of charge two SU(3) monopoles with minimal
symmetry breaking are discussed. A certain class of solutions look like SU(2)
monopoles embedded in SU(3) with a transition region or ``cloud'' surrounding
the monopoles. For large cloud size the relative moduli space metric splits as
a direct product AH\times R^4 where AH is the Atiyah-Hitchin metric for SU(2)
monopoles and R^4 has the flat metric. Thus the cloud is parametrised by R^4
which corresponds to its radius and SO(3) orientation. We solve for the
long-range fields in this region, and examine the energy density and rotational
moments of inertia. The moduli space metric for these monopoles, given by
Dancer, is also expressed in a more explicit form.Comment: 17 pages, 3 figures, latex, version appearing in Phys. Rev.

### Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras

The Perk--Schultz model may be expressed in terms of the solution of the
Yang--Baxter equation associated with the fundamental representation of the
untwisted affine extension of the general linear quantum superalgebra
$U_q[sl(m|n)]$, with a multiparametric co-product action as given by
Reshetikhin. Here we present analogous explicit expressions for solutions of
the Yang-Baxter equation associated with the fundamental representations of the
twisted and untwisted affine extensions of the orthosymplectic quantum
superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the
Perk--Schultz model.Comment: 10 pages, 2 figure

### Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(2|n)]

Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so
contains a \textit{universal $R$-matrix} in the tensor product algebra which
satisfies the Yang-Baxter equation. Applying the vector representation $\pi$,
which acts on the vector module $V$, to one side of a universal $R$-matrix
gives a Lax operator. In this paper a Lax operator is constructed for the
$C$-type quantum superalgebras $U_q[osp(2|n)]$. This can in turn be used to
find a solution to the Yang-Baxter equation acting on $V \otimes V \otimes W$
where $W$ is an arbitrary $U_q[osp(2|n)]$ module. The case $W=V$ is included
here as an example.Comment: 15 page

### 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

### 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

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