On this page we provide links to useful software for Lie algebras, algebras, groups, representation theory
and related topics. This page is part of the web-pages for the book

• CHEVIE
  CHEVIE is a computer algebra package for symbolic calculations with generic character tables of groups of Lie type, Weyl groups and Hecke algebras. It is based on the computer algebra systems GAP and Maple.

  CREP

  CREP is a package of programs to work with particular classes of finite-dimensional algebras and some combinatorial aspects of their finite-dimensional representations.

  GAP

  GAP (Groups, Algorithms, and Programming) is a system for computational discrete algebra, with particular emphasis on computational group theory. GAP contains e.g. routines for dealing with Weyl groups and Hecke algebras. For more information see the GAP Manual.

  Special routines in GAP allow to perform calculations for Weyl Groups and Hecke Algebras.

  Kac

  Kac is a C program designed to perform calculations with twisted and untwisted affine Lie algebras, including e.g. the Kac-Peterson formula for the modular matrix S and applications to coset conformal field theories. kac has most functionalities of wei, but it has no function to compute the weight system of a finite-dimensional irreducible representation of a finite-dimensional simple Lie algebra. You can download kac from its homepage. Its author is Bert Schellekens (NIKHEF, Amsterdam, The Netherlands).

  LiE

  LiE is a software system for calculations in Lie algebras. It focuses on the representation theory of complex semisimple and reductive Lie algebras (allowing e.g. for the computation of tensor product decompositions), and on the structure of their Weyl groups and root systems. It includes such features as branching rules, Bruhat ordering and Kazhdan-Lusztig polynomials.

  There is also an unofficial variant called LiEGAP which is callable from GAP.

  LIE (REDUCE package)

  LIE is also the name of a REDUCE package of functions for the classification of real n-dimensional Lie algebras.

  Magma

  Magma is designed to solve computational problems in algebra, number theory, geometry and combinatorics. Magma provides a mathematical environment for computing with algebraic and algebraic-geometric objects.

  Maple

  In the Maple Share library which is distributed with more recent releases of Maple, there are the packages coxeter and weyl. The coxeter package allows to deal with root systems, reflection groups and characters, while the weyl package contains procedures for manipulating weight vectors and computing weight multiplicities for irreducible representations of semisimple Lie algebras.

  There is an online tutorial for coxeter and weyl. The packages can be obtained by anonymous ftp. Anonymous ftp distribution sites for the Maple share library are daisy.uwaterloo.ca , ftp.inria.fr, or canc.can.nl.

  Mathematica

  There is a Mathematica program lie, available in the MathSource distribution, which allows to construct solutions of many linear and also non-linear (systems of) differential equations by making use of an underlying Lie algebra structure.

  MathLie

  MathLie is a Mathematica program which allows to determine the symmetries of (ordinary and partial, linear and nonlinear) differential equations. One can also obtain the algebraic properties of the related Lie algebra, differential representations of the Lie algebra, etc.

  Schur

  Schur is an interactive program that can be used to calculate branching rules, Kronecker products, fusion rules of WZW theories, Casimir invariants, dimensions, Young diagrams and their hook lengths etc. Its algorithms are based on the theory of partitions (which underlies the representation theory of the symmetric group), and in particular on Schur functions.

  simpLie

  simpLie is a Macintosh application that performs standard computations in simple Lie algebras and their finite dimensional representations over the complex numbers. For such representations, the software computes weight multiplicities, branching rules, tensor product decompositions etc. Version two of simpLie includes various extensions such as elements of finite order (EFO's), coset graphs and quasicrystals.

  Symmetrica

  Symmetrica is a computer algebra package devoted to representation theory, invariant theory and combinatorics of finite symmetric groups and related classes of groups.

  wei

  wei is a Fortran 77 program that computes dimensions and weight systems of irreducible representations of finite-dimensional simple Lie algebras, tensor product decompositions of such representations, and also the corresponding fusion rules of WZW theories, and quantum dimensions. To obtain it, send a mail to Jürgen Fuchs.

Other useful links can be found at the server of CAN in Amsterdam, or at the server of Penn State University.

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Last modified: 1 April 2002