Quantum and Braided Linear Algebra

Quantum matrices $A(R)$ are known for every $R$ 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, $V(R)$ (vectors) and $V^*(R)$ (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 $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $\Psi$ then their braided tensor-product V(R)\und\tens V^*(R) is a realization of the braided matrices $B(R)$ 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 $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$.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 $r$-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 $r$-matrix type Poisson orbits. Then we describe the $r$-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or $CP^n$-type orbits of $SL(n,C)$. Here we calculate symplectic leaves and the integrable foliation associated with the pencil. We also describe the algebra of functions on $CP^n$-type orbits. In Section 2 we calculate the Poisson homology of Drinfeld--Sklyanin Poisson brackets which belong to the $r$-matrix Poisson family