223 research outputs found
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.Comment: 11 pages, added 6 references and conclusing remark
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 analog 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.Comment: 28 page
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.Comment: 24 page
Energy-level crossings and number-parity effects in a bosonic tunneling model
An exactly solved bosonic tunneling model is studied along a line of the
coupling parameter space, which includes a quantum phase boundary line. The
entire energy spectrum is computed analytically, and found to exhibit multiple
energy level crossings in a region of the coupling parameter space. Several key
properties of the model are discussed, which exhibit a clear dependence on
whether the particle number is even or odd.Comment: 12 pages, 7 figure
Exactly Solvable BCS-BEC crossover Hamiltonians
We demonstrate a novel approach that allows the determination of very general
classes of exactly solvable Hamiltonians via Bethe ansatz methods. This
approach combines aspects of both the co-ordinate Bethe ansatz and algebraic
Bethe ansatz. The eigenfunctions are formulated as factorisable operators
acting on a suitable reference state. Yet, we require no prior knowledge of
transfer matrices or conserved operators. By taking a variational form for the
Hamiltonian and eigenstates we obtain general exact solvability conditions. The
procedure is conducted in the framework of Hamiltonians describing the
crossover between the low-temperature phenomena of superconductivity, in the
Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einstein condensation (BEC).Comment: 6 Pages, To appear in Proceedings of The XXIXth International
Colloquium on Group-Theoretical Methods in Physics at Chern Institute of
Mathematic
A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras
We introduce a search algorithm that utilises differential operator
realisations to find polynomial Casimir operators of Lie algebras. To
demonstrate the algorithm, we look at two classes of examples: (1) the model
filiform Lie algebras and (2) the Schr\"odinger Lie algebras. We find that an
abstract form of dimensional analysis assists us in our algorithm, and greatly
reduces the complexity of the problem.Comment: 22 page
On Casimir Operators of Conformal Galilei Algebras
In previous work, we introduced an algorithm that utilises differential
operator realisations to find polynomial Casimir operators of Lie algebras. In
this article we build on this work by applying the algorithm to several classes
of finite dimensional conformal Galilei algebras with central extension. In
these cases we highlight the utility of an algebra anti-automorphism, and give
relevant details through key examples.Comment: 18 page
Matrix elements and duality for type 2 unitary representations of the Lie superalgebra gl(m|n)
The characteristic identity formalism discussed in our recent articles is
further utilized to derive matrix elements of type 2 unitary irreducible
modules. In particular, we give matrix element formulae for all
gl(m|n) generators, including the non-elementary generators, together with
their phases on finite dimensional type 2 unitary irreducible representations.
Remarkably, we find that the type 2 unitary matrix element equations coincide
with the type 1 unitary matrix element equations for non-vanishing matrix
elements up to a phase.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1311.424
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