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SLA

Simple Lie Algebras

Version 1.5

July 2018

Willem de Graaf
Email: degraaf@science.unitn.it
Homepage: http://www.science.unitn.it/~degraaf/
Address:
Dipartimento di Matematica
Via Sommarive 14
I-38123 Povo (Trento)
Italy

Abstract

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

Copyright

© 2018 Willem de Graaf

Contents

2 Auxiliary Functions

2.1 Root Systems

  2.1-1 ExtendedCartanMatrix
  2.1-2 CartanType
  2.1-3 WeylTransversal
  2.1-4 SizeOfWeylGroup

2.2 Lie Algebras and Their Modules

  2.2-1 IsomorphismOfSemisimpleLieAlgebras
  2.2-2 AdmissibleLattice
  2.2-3 DirectSumDecomposition
  2.2-4 CharacteristicsOfStrata

3 Nilpotent Orbits

3.1 The functions

  3.1-1 NilpotentOrbit
  3.1-2 NilpotentOrbits
  3.1-3 WeightedDynkinDiagram
  3.1-4 WeightedDynkinDiagram
  3.1-5 AmbientLieAlgebra
  3.1-6 SemiSimpleType
  3.1-7 SL2Triple
  3.1-8 RandomSL2Triple
  3.1-9 SL2Grading
  3.1-10 SL2Triple
  3.1-11 InducedNilpotentOrbits
  3.1-12 RigidNilpotentOrbits

4 Finite Order Automorphisms and θ-Groups

4.1 The functions

  4.1-1 FiniteOrderInnerAutomorphisms
  4.1-2 FiniteOrderOuterAutomorphisms
  4.1-3 Order
  4.1-4 KacDiagram
  4.1-5 Grading
  4.1-6 NilpotentOrbitsOfThetaRepresentation
  4.1-7 ClosureDiagram
  4.1-8 CarrierAlgebra
  4.1-9 CartanSubspace

5 Semisimple Subalgebras of Semisimple Lie Algebras

5.1-1 ProjectionMatrix
5.1-2 Branching

5.2 Constructing Semisimple Subalgebras

  5.2-1 RegularSemisimpleSubalgebras
  5.2-2 SSSTypes
  5.2-3 LieAlgebraAndSubalgebras
  5.2-4 InclusionsGraph
  5.2-5 SubalgebrasInclusion
  5.2-6 DynkinIndex
  5.2-7 AreLinearlyEquivalentSubalgebras
  5.2-8 MakeDatabaseEntry
  5.2-9 ClosedSubsets
  5.2-10 DecompositionOfClosedSet
  5.2-11 IsSpecialClosedSet
  5.2-12 LieAlgebraOfClosedSet

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