‣ IsSimpleLieAlgebra ( L ) | ( function ) |
This function determines whether a given Lie algebra L is simple.
‣ IdealsOfLieAlgebra ( L ) | ( function ) |
This function determines the ideals of a given Lie algebra L.
‣ SimpleFactorsLieAlgebra ( L ) | ( function ) |
This function determines the simple factors of a given Lie algebra L.
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).
‣ 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.
Kaplansky [Kap82] introduced several infinite series of simple Lie algebras in characeristic two. These are available in the following functions.
‣ SimpleLieAlgebraByGramMatrix1 ( n ) | ( function ) |
For given n≥ 4 this returns a simple Lie algebra of dimension 2^n-2 (Kaplansky Type I).
‣ SimpleLieAlgebraByGramMatrix2 ( n ) | ( function ) |
For a given even n this returns a simple Lie algebra of dimension 2^n-1 (Kaplansky Type II).
‣ SimpleLieAlgebraAlternateMats ( n ) | ( function ) |
For given n this returns a simple Lie algebra of dimension n(n-1)/2 (Kaplansky Type III).
‣ 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).
generated by GAPDoc2HTML