1,510 research outputs found
Quantum and Braided Linear Algebra
Quantum matrices are known for every matrix obeying the Quantum
Yang-Baxter Equations. It is also known that these act on `vectors' given by
the corresponding Zamalodchikov algebra. We develop this interpretation in
detail, distinguishing between two forms of this algebra, (vectors) and
(covectors). A(R)\to V(R_{21})\tens V^*(R) is an algebra
homomorphism (i.e. quantum matrices are realized by the tensor product of a
quantum vector with a quantum covector), while the inner product of a quantum
covector with a quantum vector transforms as a scaler. We show that if
and are endowed with the necessary braid statistics then their
braided tensor-product V(R)\und\tens V^*(R) is a realization of the braided
matrices introduced previously, while their inner product leads to an
invariant quantum trace. Introducing braid statistics in this way leads to a
fully covariant quantum (braided) linear algebra. The braided groups obtained
from act on themselves by conjugation in a way impossible for the
quantum groups obtained from .Comment: 27 page
Spectral parameterization for the power sums of quantum supermatrix
A parameterization for the power sums of GL(m|n) type quantum (super)matrix
is obtained in terms of it's spectral values.Comment: 12 pages. Submitted to the Proceedings of the international workshop
on Classical and Quantum Integrable Systems, January 21-24, 2008, Protvino,
Russi
Absorption suppression in photonic crystals
We study electromagnetic properties of periodic composite structures, such as
photonic crystals, involving lossy components. We show that in many cases a
properly designed periodic structure can dramatically suppress the losses
associated with the absorptive component, while preserving or even enhancing
its useful functionality. As an example, we consider magnetic photonic
crystals, in which the lossy magnetic component provides nonreciprocal Faraday
rotation. We show that the electromagnetic losses in the composite structure
can be reduced by up to two orders of magnitude, compared to those of the
uniform magnetic sample made of the same lossy magnetic material. Importantly,
the dramatic absorption reduction is not a resonance effect and occurs over a
broad frequency range covering a significant portion of photonic frequency
band
Analytic model for a frictional shallow-water undular bore
We use the integrable Kaup-Boussinesq shallow water system, modified by a
small viscous term, to model the formation of an undular bore with a steady
profile. The description is made in terms of the corresponding integrable
Whitham system, also appropriately modified by friction. This is derived in
Riemann variables using a modified finite-gap integration technique for the
AKNS scheme. The Whitham system is then reduced to a simple first-order
differential equation which is integrated numerically to obtain an asymptotic
profile of the undular bore, with the local oscillatory structure described by
the periodic solution of the unperturbed Kaup-Boussinesq system. This solution
of the Whitham equations is shown to be consistent with certain jump conditions
following directly from conservation laws for the original system. A comparison
is made with the recently studied dissipationless case for the same system,
where the undular bore is unsteady.Comment: 24 page
Poisson homology of r-matrix type orbits I: example of computation
In this paper we consider the Poisson algebraic structure associated with a
classical -matrix, i.e. with a solution of the modified classical
Yang--Baxter equation. In Section 1 we recall the concept and basic facts of
the -matrix type Poisson orbits. Then we describe the -matrix Poisson
pencil (i.e the pair of compatible Poisson structures) of rank 1 or -type
orbits of . Here we calculate symplectic leaves and the integrable
foliation associated with the pencil. We also describe the algebra of functions
on -type orbits. In Section 2 we calculate the Poisson homology of
Drinfeld--Sklyanin Poisson brackets which belong to the -matrix Poisson
family
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