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# SLA

## Simple Lie Algebras

Version 0.13

March 2013

Willem de Graaf
Email: degraaf@science.unitn.it
Homepage: http://www.science.unitn.it/~degraaf/
Address:
Dipartimento di Matematica
Via Sommarive 14
I-38050 Povo (Trento)
Italy

### Abstract

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

### Copyright

© 2013 Willem de Graaf

### Contents

1 Introduction
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
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 Branching
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
References
Index
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