Lie algebras in vortex dynamics and celestial mechanics — IV

1.Classificaton of the algebra of n vortices on a plane 2.Solvable problems of vortex dynamics 3.Algebraization and reduction in a three-body problem The work [13] 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 [14,15]. 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.