Algebraic and geometric properties of quadratic Hamiltonians determined by sectional operators

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 ad _?x ^* a = ad _β ^* 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 [?x, 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 $dot x = ad_{varphi x}^* x$ .