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# Sophus

## Computing in nilpotent Lie algebras

1.24

9 April 2018

Csaba Schneider
Email: csaba@mat.ufmg.br
Homepage: http://www.mat.ufmg.br/~csaba/
Address:
Departamento de Matemática
Instituto de Ciências Exatas
Universidade Federal de Minas Gerais (UFMG)
Belo Horizonte, Brasil

The GAP Team
Email: support@gap-system.org

### Abstract

Sophus is a GAP4 package to compute with nilpotent Lie algebras over finite prime fields. In particular, the package can be used to compute certain central extensions and the automorphism group of such Lie algebras. Sophus also enables its user to test isomorphism between two nilpotent Lie algebras. The author of the package used it to construct all Lie algebras of dimension at most 9 over \(\mathbb F_2\).

### Copyright

© 2004, 2005 Csaba Schneider

The Sophus package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

### Acknowledgements

Most of the work on this package was carried out while I held a research position at the Technische Universität Braunschweig. I would like to express my gratitude to the staff and the students of the Institut für Geometrie for their interest in this work. Special thanks go to Bettina Eick for her rôle in completing this project.

### Contents

1 The theory
2 A sample calculation with Sophus
3 Sophus functions
3.1 Some general functions to compute with Lie algebras
3.1-1 SophusTest
3.1-2 IsLieNilpotentOverFp
3.1-3 MinimalGeneratorNumber
3.1-4 AbelianLieAlgebra
3.2 Functions to compute with nilpotent bases
3.2-1 NilpotentBasis
3.2-2 LieNBWeights
3.2-3 LieNBDefinitions
3.2-4 IsNilpotentBasis
3.2-5 IsLieAlgebraWithNB
3.3 The cover
3.3-1 LieCover
3.3-2 CoverHomomorphism
3.3-3 CoverOf
3.3-4 IsLieCover
3.3-5 LieMultiplicator
3.3-6 LieNucleus
3.4 Automorphisms of nilpotent Lie algebras
3.4-1 NilpotentLieAutomorphism
3.4-2 IdentityNilpotentLieAutomorphism
3.4-3 IsNilpotentLieAutomorphism
3.5 Automorphism group and isomorphism testing
3.5-1 AutomorphismGroup
3.5-2 AutomorphismGroupNilpotentLieAlgebra
3.5-3 AreIsomorphicNilpotentLieAlgebras
3.6 Descendants
3.6-1 Descendants
3.6-2 DescendantsOfStep1OfAbelianLieAlgebra
3.7 Input and output
3.7-1 WriteLieAlgebraToString
3.7-2 ReadStringToNilpotentLieAlgebra
3.7-3 WriteLieAlgebraListToFile
References
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