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Series


Series





DerivedSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form its derived
series. 



LowerCentralSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form its derived
series. 



UpperCentralSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form the upper central series of L.
The function repeatedly uses the algorithm for computing centres
while keeping track of the pre-images of the ideals factored out.





Example AlgLie_Series (H49E5)

We compute each of the type of series of a particular subalgebra of
the simple Lie algebra of type F_4 over the rational field.





> L:=SimpleLieAlgebra("F", 4, RationalField());
> L;
Lie Algebra of dimension 52 with base ring Rational Field
> K:=sub< L | [L.1, L.12, L.23, L.34, L.45] >;
> DerivedSeries(K);
[
    Lie Algebra of dimension 20 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 7 with base ring Rational Field,
    Lie Algebra of dimension 0 with base ring Rational Field
]
> LowerCentralSeries(K);
[
    Lie Algebra of dimension 20 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 12 with base ring Rational Field,
    Lie Algebra of dimension 8 with base ring Rational Field,
    Lie Algebra of dimension 5 with base ring Rational Field,
    Lie Algebra of dimension 2 with base ring Rational Field,
    Lie Algebra of dimension 1 with base ring Rational Field,
    Lie Algebra of dimension 0 with base ring Rational Field
]
> UpperCentralSeries(K);
[
    Lie Algebra of dimension 2 with base ring Rational Field,
    Lie Algebra of dimension 3 with base ring Rational Field,
    Lie Algebra of dimension 5 with base ring Rational Field,
    Lie Algebra of dimension 8 with base ring Rational Field,
    Lie Algebra of dimension 12 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 20 with base ring Rational Field
]





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