Given a Lie algebra L, return whether L is solvable.

Given a Lie algebra L, return whether L is nilpotent.

Let L be a Lie algebra over a field of characteristic p>0. Then L is called restricted if the Lie algebra ( ad) L is closed under the p-th power map (which associates to a matrix its p-th power). This function returns whether L is restricted. Lie algebras of characteristic zero are never restricted. Furthermore, it is enough to test the property for a basis of L.

Given a Lie algebra L, this function returns 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 algabra of characteristic p>0 then L need not have a Levi subalgebra but the function will always find one if one exists.

> L:=SimpleLieAlgebra("D",3,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

> M:=SimpleLieAlgebra("B",3,GF(5));
> M;
Lie Algebra of dimension 21 with base ring GF(5)
> IsRestrictedLieAlgebra( M );
true