Reflection Equation Algebra of a $(h,w)$-deformed Oscillator

Abstract

We consider the reflection equation algebra for a finite dimensional R-matrix for the $(h,w)$-deformed Heisenberg algebra ${\cal U}_{h,w}(h(4))$. A representation of the reflection matrix $K$ is constructed using the matrix generators $L^{(\pm)}$ of the ${\cal U}_{h,w}(h(4))$ algebra. A series of representations of the K-matrix then may be generated by using the coproduct rules of the ${\cal U}_{h,w}(h(4))$ algebra. The complementary condition necessary for combining two distinct solutions of the reflection equation algebra yields the braiding relations between these two sets of generators. This may be thought as a generalization of Bose-Fermi statistics to braiding statistics, which them may be used to provide a new braided colagebraic structure to a Hopf algebra generated by the elements of the matrix $K$. The reflection equation algebra and the braided exchange properties are found to depend on both deformation parameters $h$ and $w$.