LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory
Robert Feger, Thomas W. Kephart, Robert J. Saskowski
TL;DR
LieART 2.0 broadens Lie-algebra computations in Mathematica by adding extensive, reproducible branching rules to all classical and exceptional algebras up to rank 15, alongside refined tensor-product and subalgebra-decomposition capabilities. It achieves this through projection-matrix methods for subalgebras, Klimyk’s formula (and Young tableaux for SU(N)) for tensor products, and a cohesive, user-friendly interface with LaTeX output options. The paper provides a self-contained theoretical foundation, implementation details, and comprehensive benchmarks, complemented by an extensive appendix of maximal-subalgebra data and supplementary tables. This work significantly enhances model-building workflows in theoretical physics and related areas by delivering a robust, scalable tool for representation-theory calculations and their applications.
Abstract
We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged: it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.