Phillip S. Isaac

Weak Hopf algebras corresponding to Uq[sln]

We investigate the weak Hopf algebras of Li based on Uq[sln] and Sweedler’s finite dimensional example. We give weak Hopf algebra isomorphisms between the weak generalizations of Uq[sln] which are “upgraded” automorphisms of Uq[sln] and hence give a classification of these structures as weak Hopf algebras. We also show how to decompose these examples into a direct sum which leads to unexpected isomorphisms between their algebraic structure.

Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrodinger algebra.

On irreducible representations of the exotic conformal Galilei algebra

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite-dimensional irreducible modules is presented.

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analogue of this construction for the most general polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

Invariants and reduced matrix elements associated with the Lie superalgebra gl(m|n)

We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix elements of generators, and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra.

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 two-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

An integrable case of the p + ip pairing Hamiltonian interacting with its environment

We consider a generalization of the p + ip pairing Hamiltonian, with external interaction terms of a particular form. These terms allow for the exchange of particles between the system and its environment. As a result the u(1) symmetry associated with conservation of particle number, present in the p + ip Hamiltonian, is broken. Nonetheless the generalized model is integrable. We establish integrability using the boundary quantum inverse scattering method, with one of the reflection matrices chosen to be non-diagonal. We also derive the corresponding Bethe ansatz equations, the roots of which parametrize the exact solution for the energy spectrum.

A variational approach for the quantum inverse scattering method

We introduce a variational approach for the quantum inverse scattering method to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake this in a manner which does not rely on any prior knowledge of integrability through the existence of a set of conserved operators. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen–Cooper–Schrieffer theory, and Bose–Einstein condensation. The Hamiltonians considered describe systems with interacting Cooper pairs and a bosonic degree of freedom. We obtain general exact solvability requirements which include seven subcases that have previously appeared in the literature.

BEC-BCS crossover in a (p + ip)-wave pairing Hamiltonian coupled to bosonic molecular pairs

We analyse a (p + ip)-wave pairing BCS Hamiltonian, coupled to a single bosonic degree of freedom representing a.molecular condensate, and investigate the nature of the BEC-BCS crossover for this system. For a suitable restriction on the coupling parameters, we show that the model is integrable and we derive the exact solution by the algebraic Bethe ansatz. In this manner we also obtain explicit formulae for correlation functions and compute these for several cases. We find that the crossover between the BEC state and the strong pairing p + ip phase is smooth for this model, with no intermediate quantum phase transition. Crown Copyright (C) 2011 Published by Elsevier B.V. All rights reserved.

Assessment of an Innovative System of Lecture Notes in First-Year Mathematics

Lectures are a familiar component in the delivery of mathematical content. Lecturers are often challenged with presenting material in a manner that aligns with the various learning styles and abilities within a large class. Students complain that the old-fashioned lecture style of copying notes from a board hinders the learning process, as they simply concentrate on writing. In recent times, distributing elaborate lecture notes has become a widespread alternative, but has its own problems, alienating the audience with lack of participation. The authors have developed a system of lecture notes, we call partially populated lecture notes, that have enjoyed success with students and addressed these difficulties.