Boyko V. Generalized Casimir operators, singular reduction modules and symmetries of differential equations

In the thesis, the main attention is paid to problems related to generalized Casimir operators and realizations of Lie algebras as well as Lie symmetries and reduction modules of differential equations. An original algorithm for finding fundamental bases of invariants (generalized Casimir operators) of Lie algebras is developed, which uses the Cartan method of moving frames in the Fels-Olver version. In order to test the method, demonstrate its advantages and explain the normalization procedure, invariants of real low-dimensional Lie algebras are re-computed. We completely describe for the first time invariant bases for series of solvable Lie algebras of arbitrary dimension with fixed structures of nilradicals. A rigorous definition of reduction modules of differential equations is introduced, which allows us to revisit the theory of nonclassical sym