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SLA

Computing with simple Lie algebras

Version 1.6.2

12 July 2024

Willem Adriaan de Graaf
Email: degraaf@science.unitn.it
Homepage: http://www.science.unitn.it/~degraaf

Abstract

This package provides functions for computing with various aspects of the theory of simple Lie algebras in characteristic zero.

Copyright

© 2013-2018 Willem de Graaf

Contents

2 Root Systems and Weyl Groups

2.1 Root Systems

  2.1-1 ExtendedCartanMatrix
  2.1-2 CartanType
  2.1-3 DisplayDynkinDiagram

2.2 Weyl groups

  2.2-1 WeylTransversal
  2.2-2 SizeOfWeylGroup
  2.2-3 WeylGroupAsPermGroup
  2.2-4 ApplyWeylPermToWeight
  2.2-5 WeylWordAsPerm
  2.2-6 PermAsWeylWord

3 Semisimple Lie Algebras and their Modules

3.1 Semisimple Lie algebras

  3.1-1 IsomorphismOfSemisimpleLieAlgebras
  3.1-2 DisplayDynkinDiagram
  3.1-3 ApplyWeylPermToCartanElement

3.2 Representations of semisimple Lie algebras

  3.2-1 AdmissibleLattice
  3.2-2 DirectSumDecomposition
  3.2-3 IsIrreducibleHWModule
  3.2-4 HighestWeightVector
  3.2-5 HighestWeight
  3.2-6 DisplayHighestWeight
  3.2-7 IsomorphismOfIrreducibleHWModules
  3.2-8 DualAlgebraModule
  3.2-9 CharacteristicsOfStrata

5 Finite Order Automorphisms and \(\theta\)-Groups

5.1 The functions

  5.1-1 FiniteOrderInnerAutomorphisms
  5.1-2 FiniteOrderOuterAutomorphisms
  5.1-3 Order
  5.1-4 KacDiagram
  5.1-5 Grading
  5.1-6 NilpotentOrbitsOfThetaRepresentation
  5.1-7 ClosureDiagram
  5.1-8 CarrierAlgebra
  5.1-9 CartanSubspace

6 Semisimple Subalgebras of Semisimple Lie Algebras

6.1-1 ProjectionMatrix
6.1-2 Branching

6.2 Constructing Semisimple Subalgebras

  6.2-1 RegularSemisimpleSubalgebras
  6.2-2 SSSTypes
  6.2-3 LieAlgebraAndSubalgebras
  6.2-4 InclusionsGraph
  6.2-5 SubalgebrasInclusion
  6.2-6 DynkinIndex
  6.2-7 AreLinearlyEquivalentSubalgebras
  6.2-8 MakeDatabaseEntry
  6.2-9 ClosedSubsets
  6.2-10 DecompositionOfClosedSet
  6.2-11 IsSpecialClosedSet
  6.2-12 LieAlgebraOfClosedSet

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