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 {x1, ... , xn} is a basis of L, return the Killing matrixof L, which is defined to be the matrix ((Tr) ((ad)xi.(ad) xj)).

Example AlgLie_Other (H107E43)

> 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]
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], Para 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 AlgLie_Predicates (H107E44)

We test various predicates in the context of the simple Lie algebra of type D3 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
V2.28, 13 July 2023