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Operations on Lie Algebras
Operations on Lie Algebras
Centre(L) : AlgLie -> AlgLie
Given a Lie algebra L, return the centre of L.
Centralizer(L, K) : AlgLie, AlgLie -> AlgLie
Given a Lie algebra L and a subalgebra K of L,
return the centralizer of K in L.
Normalizer(L, K) : AlgLie, AlgLie -> AlgLie
Given a Lie algebra L and a subalgebra K of L,
return the normalizr of K in L.
SolvableRadical(L) : AlgLie -> AlgLie
Given a Lie algebra L, return the solvable radical of L.
NilRadical(L) : AlgLie -> AlgLie
Given a Lie algebra L, return the nilradical of L.
Example AlgLie_Operations (H49E3)
We demonstrate the functions for performing basic operations with Lie algebras
like centre, etc.
> L := SimpleLieAlgebra("D", 4, RationalField());
> L;
Lie Algebra of dimension 28 with base ring Rational Field
> Centre(L);
Lie Algebra of dimension 0 with base ring Rational Field
> K := sub< L | [L.1, L.2, L.3] >;
> Centralizer(L, K);
Lie Algebra of dimension 10 with base ring Rational Field
> Normalizer(L, K);
Lie Algebra of dimension 19 with base ring Rational Field
> M := Centralizer(L, K);
> S := SolvableRadical(M);
> S;
Lie Algebra of dimension 10 with base ring Rational Field
> Morphism(S, L);
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
> NilRadical(M);
Lie Algebra of dimension 9 with base ring Rational Field
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