The Poisson bracket compatible with the classical reflection equation algebra

Abstract

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.

13 pages, LaTeX with AmsFonts

References in corpus (3)

• Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices
• A family of the Poisson brackets compatible with the Sklyanin bracket
• New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-hamiltonian structure

Cited by in corpus (8)

• On maximally superintegrable systems
• New variables of separation for particular case of the Kowalevski top
• On bi-hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion
• On natural Poisson bivectors on the sphere
• On the generalized integrable Chaplygin system
• On bi-integrable natural Hamiltonian systems on the Riemannian manifolds
• On bi-hamiltonian structure of some integrable systems on so*(4)
• On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$