Algebraic and geometric properties of quadratic Hamiltonians determined by sectional operators

Abstract

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator \(\varphi :\mathfrak{g}* \to \mathfrak{g}\) is sectional if it satisfies the identity adϕx*a = adβ*x, \(x \in \mathfrak{g}*\), where \(\mathfrak{g}\) is a finite-dimensional Lie algebra and \(a \in \mathfrak{g}*\) and \(\beta \in \mathfrak{g}\) are fixed elements. In the case of a semisimple Lie algebra \(\mathfrak{g}\), the above identity takes the form [ϕx, a] = [β, x] and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of n-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation \(\dot x = ad_{\varphi x}^* x\).