Maximally superintegrable Gaudin magnet: A unified approach RID A-7283-2010

A classical integrable Hamiltonian system is defined by an A belian subalgebra (of suitable dimension) of a Poisson algebra, while a quantum integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Jordan-Lie algebra of Hermitian operators. We propose a method for obtaining "large." Abelian subalgebras inside the tensor product of free tensor algebras, and we show, that there exist canonical morphisms from these algebras to Poisson algebras and Jordan-Lie algebras of operators. We can thus prove the integrability of some particular Hamiltonian systems simultaneously at both the classical and the quantum level. We propose a particular case of the rational Gaudin magnet as all example.