We introduce a family of compatible Poisson brackets on the space of $2 \times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.
Keywords:
Poisson bracket, bi-hamiltonian structure, reflection equation algebra
Citation:
Tsiganov A. V., The Poisson Bracket Compatible with the Classical Reflection Equation Algebra, Regular and Chaotic Dynamics,
2008, Volume 13, Number 3,
pp. 191-203