
Journal of Lie Theory 30 (2020), No. 2, 425444 Copyright Heldermann Verlag 2020 AffineQuadratic Problems on Lie Groups: Tops and Integrable Systems Velimir Jurdjevic Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada jurdj@math.toronto.edu [Abstractpdf] \newcommand{\fp}{\mathfrak{p}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\fk}{\mathfrak{k}} This paper focuses on the relevance of a certain class of leftinvariant Hamiltonians (affinequadratic) on a reductive semisimple Lie algebra $\fg$ for the theory of integrable systems and the equations of applied mathematics. Any semisimple Lie group $G$ that contains a closed subgroup $K$ is reductive, in the sense, that the orthogonal complement $\fp$ in $\fg$ of the Lie algebra $\fk$ of $K$, relative to the Killing form, satisfies $[\fk,\fp]\subseteq \fp$. This implies that $K$ acts (by adjoint action) on $\fp$ and therefore induces the semidirect product $\fp\rtimes K$. Consequently, $\fg$, as a vector space, carries two Lie algebra structures: semisimple, and semidirect. Hence, the dual $\fg^*$ carries two Poisson structures as well. Any affinequadratic function $H$ on $\fg$ can be simultaneously viewed as a Hamiltonian for either Poisson structure.\\[2mm] We will show that certain coadjoint orbits relative to the semidirect action are the cotangent bundles of $SO(n)$. This implies that the equations of an $n$dimensional top can be represented on such coadjoint orbits. In this situation there is a canonical affinequadratic Hamiltonian whose Hamiltonian equations on these coadjoint orbits coincide with the equations of the top. This implies that the integrable cases of the top correspond to the the integrable cases of the overseeing affine Hamiltonian. More generally, we will identify a subclass of affinequadratic Hamiltonians, called isospectral, that provides new insights into the theory of integrable systems based on the contributions of S.\,V.\, Manakov, A.\,T.\,Fomenko, A.\,S.\,Mischenko and O.\,Bogoyavlensky listed in the references. Keywords: Lie groups, symplectic manifolds, Poisson manifolds, control systems, Hamiltonian systems, maximum principle. MSC: 70H45, 70H05; 57R27, 53C22. [ Fulltextpdf (172 KB)] for subscribers only. 