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We suggest a programming realization of an algorithm for verifying a givenset of algebraic relations in the form of a supercommutator multiplicationtable for the Verma module, which is constructed according to a generalizedCartan procedure for a quadratic superalgebra and whose elements are realizedas a formal power series with respect to non-commuting elements. To this end,we propose an algebraic procedure of Verma module construction and itsrealization in terms of non-commuting creation and annihilation operators of agiven Heisenberg--Weyl superalgebra. In doing so, we set up a problem whichnaturally arises within a Lagrangian description of higher-spin fields inanti-de-Sitter (AdS) spaces: to verify the fact that the resulting Verma moduleelements obey the given commutator multiplication for the original non-linearsuperalgebra. The problem setting is based on a restricted principle ofmathematical induction, in powers of inverse squared radius of the AdS-space.For a construction of an algorithm resolving this problem, we use a two-leveldata model within the object-oriented approach, which is realized on a basis ofthe programming language C#. The program allows one to consider objects (of aless general nature than non-linear commutator superalgebras) that fall underthe class of so-called $GR$-algebras, for whose treatment one widely uses themodule \emph{Plural} of the system \emph{Singular} of symbolic computations forpolynomials.

Programming Realization of Symbolic Computations for Non-linear  Commutator Superalgebras over the Heisenberg--Weyl Superalgebra: Data  Structures and Processing Methods