Nesterenko M. Realizations and contractions of Lie algebras, orbit-functions and quasicrystals

The thesis is devoted to contractions and realizations of Lie algebras, to Fourier analysis of almost periodic functions and investigation of orbit-functions and orthogonal polynomials. Two new necessary conditions of contraction existence, are proposed and applied to to the complex nilpotent Lie algebras of dimensions five and six. Zaitsev's hypothesis is revised in terms of S-expansions and it is shown that expansion and reduction allow one to obtain a non-unimodular Lie algebra from a unimodular one. A reduction theorem that transforms realization to the essential variables is proven and the Shirokov's algebraic method is generalized. A construction of the parameterized realization that converges to a contracted Lie algebra realization is proposed. Realizations and deformations of the Galilei, Poincare and conformal algebras are constructed. Complete set of differential invariants for the smallest Galilei algebra is constructed and the theorem on the normal form of an invariant system is proven. A method for the discrete Fourier analysis of almost periodic functions defined on quasicrystals is developed. The proposed method is tested using two almost periodic functions on the Fibonacci quasicrystal. Pertinent properties of the orbit-functions are studied. Three methods for the construction of orthogonal polynomials related to the orbit functions are proposed. The equivalence of the constructed polynomials to several known families of polynomials is shown.