CoReLG

Computing with real Lie algebras

Version 1.57

7 July 2024

Paolo Faccin
Email: paolofaccin86@gmail.com
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 real simple Lie algebras.

Acknowledgements

The research leading to this package has received funding from the European Union's Seventh Framework Program FP7/2007-2013 under grant agreement no 271712, and from the Australian Research Council, grantor code DE140100088 and DP190100317.

1 Introduction

1.1 The simple real Lie algebras

1.2 Cartan subalgebras and more

1.3 Nilpotent orbits

1.4 On base fields

2 The field SqrtField

2.1 Definition of the field

2.1-1 SqrtFieldIsGaussRat

2.2 Construction of elements

  2.2-1 Sqroot
  2.2-2 CoefficientsOfSqrtFieldElt
  2.2-3 SqrtFieldEltByCoefficients
  2.2-4 SqrtFieldEltToCyclotomic
  2.2-5 SqrtFieldEltByCyclotomic

2.3 Basic operations

  2.3-1 SqrtFieldMakeRational
  2.3-2 SqrtFieldPolynomialToRationalPolynomial
  2.3-3 SqrtFieldRationalPolynomialToSqrtFieldPolynomial
  2.3-4 Factors

3 Real Lie Algebras

3.1 Construction of simple real Lie algebras

  3.1-1 RealFormsInformation
  3.1-2 NumberRealForms
  3.1-3 RealFormById
  3.1-4 IdRealForm
  3.1-5 NameRealForm
  3.1-6 AllRealForms
  3.1-7 RealFormParameters
  3.1-8 IsRealFormOfInnerType
  3.1-9 IsRealification
  3.1-10 CartanDecomposition
  3.1-11 RealStructure

3.2 Maximal reductive subalgebras

3.2-1 MaximalReductiveSubalgebras

3.3 Isomorphisms

3.3-1 IsomorphismOfRealSemisimpleLieAlgebras

3.4 Cartan subalgebras and root systems

  3.4-1 MaximallyCompactCartanSubalgebra
  3.4-2 MaximallyNonCompactCartanSubalgebra
  3.4-3 CompactDimensionOfCartanSubalgebra
  3.4-4 CartanSubalgebrasOfRealForm
  3.4-5 CartanSubspace
  3.4-6 RootsystemOfCartanSubalgebra
  3.4-7 ChevalleyBasis

3.5 Diagrams

3.5-1 VoganDiagram
3.5-2 SatakeDiagram

4 Real nilpotent orbits

4.1 Nilpotent orbits in real Lie algebras

  4.1-1 NilpotentOrbitsOfRealForm
  4.1-2 RealCayleyTriple
  4.1-3 WeightedDynkinDiagram

4.2 The real Weyl group

4.2-1 RealWeylGroup

References

Index

CoReLG

Computing with real Lie algebras

Abstract

Copyright

Acknowledgements

Contents