Algebraic aspects of compatible poisson structures

2012-07-25T12:15:10Z (GMT) by Pumei Zhang
This thesis consists of three chapters. In Chapter one, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Brief statements of several open problems related to our main results are also mentioned in this part. In Chapter two, we applied the so-called Jordan–Kronecker decomposition theorem to study algebraic properties of the pencil P generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group GP that preserves P. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope GP also plays a fundamental role. In Chapter three, we study one of the famous Poisson pencils which is sometimes called “argument shift pencil”. This pencil is defined on the dual space g ∗ of an arbitrary Lie algebra g. This pencil is generated by the Lie-Poisson bracket { , } and constant bracket { , }a for a ∈ g ∗ . Thus we may apply the Jordan–Kronecker decomposition theorem to introduce the so-called Jordan–Kronecker invariants of a finite-dimensional Lie algebra g. These invariants can be understood as the algebraic type of the canonical Jordan–Kronecker form for the “argument shift pencil” at a generic point. Jordan–Kronecker invariants are found for all low-dimensional Lie algebras (dim g ≤ 5) and can be used to construct the families of polynomials in bi-involution. The results are found to be useful in the discussion of the existence of a complete family of polynomials in bi-involution w.r.t. these two brackets { , } and { , }a.