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

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