Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

Abstract

In this paper we present Affine.m—a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summaryProgram title: Affine.mCatalogue identifier: AENA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, UKLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 24844No. of bytes in distributed program, including test data, etc.: 1045908Distribution format: tar.gzProgramming language: Mathematica.Computer: i386–i686, x86_64.Operating system: Linux, Windows, Mac OS, Solaris.RAM: 5–500 MbClassification: 4.2, 5.Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems.Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases.Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical.Unusual features:We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface.Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.