Lie algebras in vortex dynamics and celestial mechanics - IV

The work of A.V. Borisov, A.E. Pavlov, Dynamics and Statics of Vortices on a Plane and a Sphere - I (Reg. & Ch. Dynamics, 1998, Vol. 3, No 1, p.28-39) introduces a naive description of dynamics of point vortices on a plane in terms of variables of distances and areas which generate Lie-Poisson structure. Using this approach a qualitative description of dynamics of point vortices on a plane and a sphere is obtained in the works Dynamics of Three Vortices on a Plane and a Sphere - II. General compact case by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No 2, p.99-114), Dynamics of three vortices on a plane and a sphere - III. Noncompact case. Problem of collaps and scattering by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No 4, p.76-90). In this paper we consider more formal constructions of the general problem of n vortices on a plane and a sphere. The developed methods of algebraization are also applied to the classical problem of the reduction in the three-body problem.