LieART -- A Mathematica Application for Lie Algebras and Representation Theory

TL;DR

LieART provides a comprehensive Mathematica-based toolkit for Lie-algebra and representation-theory computations, unifying input via dimensional names with internal Dynkin-label representations. It employs Weyl-group reflections for weight systems, Klimyk’s formula for efficient tensor-product decompositions, and projection matrices for subalgebra branching, with Young Tableaux offered as an alternative for SU(N) cases. The package delivers extensive tables, a LaTeX integration, and a user-focused interface intended for particle-physics model-building and GUT studies, while outlining benchmarks that highlight performance trade-offs, especially for exceptional algebras. Overall, LieART advances reproducible, scalable Lie-algebra computations within a widely used CAS, enabling both routine tasks and deeper explorations of maximal-subalgebra embeddings and branching patterns.

Abstract

We present 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. LieART can handle all classical and exceptional Lie algebras. It computes root systems of Lie algebras, weight systems and several other properties of irreducible representations. LieART's user interface has been created with a strong focus on usability and thus allows the input of irreducible representations via their dimensional name, while the output is in the textbook style used in most particle-physics publications. The unique Dynkin labels of irreducible representations are used internally and can also be used for input and output. LieART exploits the Weyl reflection group for most of the calculations, resulting in fast computations and a low memory consumption. Extensive tables of properties, tensor products and branching rules of irreducible representations are included in the appendix.

Figures (2)

• Figure 1: Dynkin Diagrams of classical and exceptional simple Lie algebras.
• Figure 2: Extended Dynkin Diagrams of classical and exceptional simple Lie algebras.