Chapovskyi Y. Nilpotent and solvable Lie algebras of derivations of polynomial rings

The dissertation is devoted to studying nilpotent and solvable subalgebras of the Lie algebra of derivations of associative commutative ring over a field of characteristic 0. Lie algebras of derivations are a fundamental object and find applications in numerous branches of mathematics and physics, in particular, in differential geometry, theory of ordinary differential equations, theory of differential equations with partial differential equations, algebraic geometry, theoretical physics, etc. Several examples related to the symmetric analysis of differential equations are particularly noteworthy. Symmetry analysis of differential equations — a science that studies the symmetries of differential equations and uses their properties to solve important questions about the equation itself. For example, if an ordinary differential equation of order ? has a solvable Lie algebra of symmetries of dimension ?, then knowing such a Lie algebra of symmetries the equation can be solved in quadrature. Using a similar idea, ordinary differential equations of the second order that can be solved in this way have been classified and their classification was reduced to the classification of two-dimensional Lie algebras of vector fields on the plane. For partial differential equations, the apparatus of symmetric analysis allows us to find families of special partial solutions, by knowing some Lie algebra of symmetries of such equations, and as we know finding exact solutions of partial differential equations is, generally speaking, problematic. More precisely, by considering some subalgebra of symmetries of the differential equation with partial differential equations, we can set out to find solutions that are invariant with respect to this Lie subalgebra of symmetries. This allows us to make a reduction, i.e. move to a new system of partial differential equations, in which the number of independent variables is less by an order of magnitude of the subalgebra of the Lie symmetries.