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# CoReLG

## Computing with real Lie algebras

Version 1.56

24 February 2022

Heiko Dietrich
Email: heiko.dietrich@monash.edu
Homepage: http://users.monash.edu.au/~heikod/
Address:
School of Mathematics
Monash University
Wellington Road 1
VIC 3800, Melbourne, Australia

Paolo Faccin
Email: paolofaccin86@gmail.com
Address:
Dipartimento di Matematica
Via Sommarive 14
I-38050 Povo (Trento), Italy

Willem de Graaf
Email: degraaf@science.unitn.it
Homepage: https://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 real simple Lie algebras.

### Copyright

© 2013-2019 Heiko Dietrich, Paolo Faccin, and Willem de Graaf

### 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.

### Contents

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
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