Symmetry and quaternionic integrable systems

Abstract

Abstract. Given a hyperkahler manifold M , the hyperkahler structure defines a triple of symplectic structures on M ; with these, a triple of Hamiltonians defines a so called hyperhamiltonian dynamical system on M . These systems are integrable when can be mapped to a system of quaternionic oscillators. We discuss the symmetry of integrable hyperhamiltonian systems, i.e. quaternionic oscillators; and conversely how these symmetries characterize, at least in the Euclidean case, integrable hyperhamiltonian systems.

Geometry of Riccati equations over normed division algebras

“…Particular cases of quaternionic Riccati equations, e.g. linear ones, appear in [28]. As Riccati equations over R and C have quadratic terms, it is reasonable that their generalization to quaternions should also content such a term.…”