Sophus

Computing in nilpotent Lie algebras

Version 1.23

February 2006

Csaba Schneider
e-mail: csaba.schneider@sztaki.hu
WWW: http://www.sztaki.hu/~schneider
Address:
Informatics Laboratory
Computer and Automation Research Institute
The Hungarian Academy of Sciences
1111 Budapest, L\'agym\'anyosi u.\ 11.
Hungary

Abstract

Sophus is a GAP~4 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

(C) 2004, 2005 Csaba Schneider

Acknowledgements

Most of the work on this package was carried out while I held a research position at the Technische Universtit\"at Braunschweig. I would like to express my gratitude to the staff and the students of the Institut f\"ur Geometrie for their interest in this work. Special thanks go to Bettina Eick for her r\^ole 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
      3.7-4 SophusBuildManual
      3.7-5 SophusBuildManualHTML




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