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 a verification of 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 noncommuting 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 first level, the so-called basic model of superalgebra, describes a set of operations to be realized as symbolic computations for arbitrary finite-dimensional associative superalgebras. The second level serves to realize a specific representation of non-linear commutator superalgebra elements, and specifies the peculiarities of commutation operations for the elements of a specific superalgebra A, as well as the ordering of creation f+, bi and annihilation f, bi, i = 1, 2, operators in products which determine supercommutators [a, b}, a, b ∈ A, to be verified. 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 Plural of the system Singular of symbolic computations for polynomials. [email protected] † [email protected]