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1 Introduction
 1.1 Tables
 1.2 Constructions

1 Introduction

This package contains methods to compute with finite Lie algebras; that is, with Lie algebras over finite fields. The functions in this package are often particularly effective if the considered Lie algebras are solvable, since there are various special methods for finite solvable Lie algebras implemented. The GAP system contains various methods to compute with Lie algebras, see [DeG00] for background. This package is based on these methods and extends them in the case of a finite Lie algebra, see [Eic04] for a description of some of the methods. The package also contains a database of small-dimensional simple Lie algebras over the field with two elements, see [Eic10] for additional information.

1.1 Tables

Computations with Lie algebras in FinLie are usually carried out using tables. These tables wrt. to a basis B of L contain the structure constants for the Lie algebra L. The following functions allow the computation of such a table and the construction of a Lie algebra from a given table.

1.1-1 LieTableByBasis
‣ LieTableByBasis( L, B )( function )

Determines a table for the given Lie algebra L wrt. to the given basis B.

1.1-2 LieAlgebraByTable
‣ LieAlgebraByTable( T )( function )

Constructs a Lie algebra described by the given table T.

1.2 Constructions

There are several constructions of Lie algebras implemented in FinLie; in particular the following constructions are available.

1.2-1 AbelianLieAlgebra
‣ AbelianLieAlgebra( n, p )( function )

Constructs the n-dimensional abelian Lie algebra L over the field with p elements.

1.2-2 UnipotentLieAlgebra
‣ UnipotentLieAlgebra( n, p )( function )

Constructs the unipotent Lie algebra L of n×n matrices over the field with p elements.

1.2-3 TriangularLieAlgebra
‣ TriangularLieAlgebra( n, p )( function )

Constructs the triangular Lie algebra L of n×n matrices over the field with p elements.

1.2-4 LieAlgebraTensor
‣ LieAlgebraTensor( L, K )( function )

Constructs the tensor product of a given Lie algebra L with a finite field K.

1.2-5 DerivationsLieAlgebra
‣ DerivationsLieAlgebra( L )( function )

Constructs the Lie algebra of derivations a given Lie algebra L.

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