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4 Simple Lie algebras
 4.1 Simple Lie algebras and ideals
 4.2 Small-dimensional simple Lie algebras over GF(2)
 4.3 Infinite families of simple Lie algebras over GF(2)

4 Simple Lie algebras

4.1 Simple Lie algebras and ideals

4.1-1 IsSimpleLieAlgebra
‣ IsSimpleLieAlgebra( L )( function )

This function determines whether a given Lie algebra L is simple.

4.1-2 IdealsOfLieAlgebra
‣ IdealsOfLieAlgebra( L )( function )

This function determines the ideals of a given Lie algebra L.

4.1-3 SimpleFactorsLieAlgebra
‣ SimpleFactorsLieAlgebra( L )( function )

This function determines the simple factors of a given Lie algebra L.

4.2 Small-dimensional simple Lie algebras over GF(2)

FinLie contains a database of small-dimensional simple Lie algebras over the field with two elements. This database provides a complete list of isomorphism type representatives up to dimension 9, see [VL06] for the classification. In dimensions 10 to 20 the database contains non-isomorphic simple Lie algebras, but it might not be complete. These are either obtained by the method described in [Eic10] or are contained in one of the known infinite families of simple Lie algebras over GF(2). These infinite series are also available in FinLie (see next section).

4.2-1 LieAlgebraByLibrary
‣ LieAlgebraByLibrary( p, d, n )( function )

This function returns the n-th Lie algebra in the database of d-dimensional simple Lie algebras over GF(p). Note that currently on p=2 is supported.

4.3 Infinite families of simple Lie algebras over GF(2)

Kaplansky [Kap82] introduced several infinite series of simple Lie algebras in characeristic two. These are available in the following functions.

4.3-1 SimpleLieAlgebraByGramMatrix1
‣ SimpleLieAlgebraByGramMatrix1( n )( function )

For given n≥ 4 this returns a simple Lie algebra of dimension 2^n-2 (Kaplansky Type I).

4.3-2 SimpleLieAlgebraByGramMatrix2
‣ SimpleLieAlgebraByGramMatrix2( n )( function )

For a given even n this returns a simple Lie algebra of dimension 2^n-1 (Kaplansky Type II).

4.3-3 SimpleLieAlgebraAlternateMats
‣ SimpleLieAlgebraAlternateMats( n )( function )

For given n this returns a simple Lie algebra of dimension n(n-1)/2 (Kaplansky Type III).

4.3-4 SimpleLieAlgebraByQuadraticForm
‣ SimpleLieAlgebraByQuadraticForm( Q )( function )

Let m≥ 3. Given Q a nonsingular quadratic form on a 2m-dimensional vector space V, this function returns a simple Lie algebra of dimension 2^m-1(2^m-1) if the Arf invariant of Q is 0 or 2^m-1(2^m+1) if the Arf invariant of Q is 1 (Kaplansky Type IV).

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