Properties of Lie Algebras and Ideals

KillingForm(L): AlgLie -> ModMatFldElt
KillingForm(L): AlgMatLie -> ModMatFldElt
KillingMatrix(L): AlgLie -> ModMatFldElt
KillingMatrix(L): AlgMatLie -> ModMatFldElt: Given a Lie algebra \(L\) such that \(\{x_1,\ldots ,x_n\}\) is a basis of \(L\), return the Killing matrix of \(L\), which is defined to be the matrix \(({\rm Tr} ({\rm ad}x_i\cdot {\rm ad} x_j))\).

Example: Killing Matrix of a Lie Algebra (ex-53bd1a)

> L:=LieAlgebra("B2",RationalField());
> KillingMatrix(L);
[ 0  0  0  0  0  0  0  0  0  6]
[ 0  0  0  0  0  0  0  0 12  0]
[ 0  0  0  0  0  0  0 12  0  0]
[ 0  0  0  0  0  0  6  0  0  0]
[ 0  0  0  0  6  6  0  0  0  0]
[ 0  0  0  0  6 12  0  0  0  0]
[ 0  0  0  6  0  0  0  0  0  0]
[ 0  0 12  0  0  0  0  0  0  0]
[ 0 12  0  0  0  0  0  0  0  0]
[ 6  0  0  0  0  0  0  0  0  0]

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IsAbelian(L): AlgLie -> BoolElt
IsAbelian(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is abelian.

IsSoluble(L): AlgLie -> BoolElt
IsSoluble(L): AlgMatLie -> BoolElt
IsSolvable(L): AlgLie -> BoolElt
IsSolvable(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is soluble.

IsNilpotent(L): AlgLie -> BoolElt
IsNilpotent(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is nilpotent.

IsCentral(L, M): AlgLie, AlgLie -> BoolElt
IsCentral(L, M): AlgMatLie, AlgMatLie -> BoolElt: Given a subalgebra \(M\) of the Lie algebra \(L\), return true if \(M\) is central in \(L\).

IsSimple(L): AlgLie -> BoolElt
IsSimple(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is simple.

IsSemisimple(L): AlgLie -> BoolElt
IsSemisimple(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is semisimple.

IsReductive(L): AlgLie -> BoolElt
IsReductive(L): AlgMatLie -> BoolElt: Given a Lie algebra \(L\), return true if \(L\) is reductive.

HasLeviSubalgebra(L): AlgLie -> BoolElt

HasLeviSubalgebra(L): AlgMatLie -> BoolElt

Given a Lie algebra \(L\), this function determines whether \(L\) has a Levi subalgebra. If the result is true, then the function also returns a semisimple subalgebra (complement to the solvable radical) of \(L\). If \(L\) is defined over a field of characteristic 0, then it always has a Levi subalgebra. However, if \(L\) is a Lie algebra of characteristic \(p>0\) then \(L\) need not have a Levi subalgebra but the function will always find one if it exists.

A description of the algorithm used is contained in [dG00], §4.13.

IsClassicalType(L): AlgLie -> BoolElt: Determines if the reductive Lie algebra \(L\) is of classical-type. Note that all reductive Lie algebras over fields of characteristic 0 are considered to be classical-type.

Example: Predicates on a Simple Lie Algebra (ex-6ba33a)

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We test various predicates in the context of the simple Lie algebra of type \(D_3\) over the rational field.

> L:=LieAlgebra("D3",RationalField());
> L;
Lie Algebra of dimension 15 with base ring Rational Field
> K:=sub< L | [L.1,L.2,L.3] >;
> M:=Centralizer(L, K);
> M;
Lie Algebra of dimension 4 with base ring Rational Field
> R:=SolvableRadical(M);
> R;
Lie Algebra of dimension 4 with base ring Rational Field
> HasLeviSubalgebra(M);
true Lie Algebra of dimension 0 with base ring Rational Field
> K:=Centralizer(L, sub< L | [L.1,L.2,L.3] >);
> K;
Lie Algebra of dimension 4 with base ring Rational Field
> IsSolvable(K);
true
> IsNilpotent(K);
true
> R:= SolvableRadical(K);
> IsSolvable(R);
true
> IsNilpotent(R);
true
> N:= Nilradical(K);
> IsNilpotent(N);
true

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Properties of Lie Algebras and Ideals#

Properties of Lie Algebras and Ideals