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x + y : AlgLieElt, AlgLieElt -> AlgLieElt
x + y : AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt
x - y : AlgLieElt, AlgLieElt -> AlgLieElt
x - y : AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt
x * y : AlgLieElt, AlgLieElt -> AlgLieElt
x * y : AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt
IsCentral(L, M) : AlgMatLie,AlgMatLieElt -> BoolElt
Given an element x of the Lie algebra L,
return true if x is central in L.
Given a (structure constant) Lie algebra L,
this function returns an element of L that is not nilpotent, or
the zero element of L if no such element exists.
The algorithm follows [dG00], Para 2.7.
We construct a non-nilpotent element of a Lie algebra.
> L:=LieAlgebra("G2",RationalField());
> NonNilpotentElement(L);
(0 0 0 0 0 1 0 0 0 0 0 0 0 0)
RightAdjointMatrix(L, x) : AlgLie, AlgLieElt -> AlgMatLieElt
Given a (structure constant) Lie algebra L and an element x of a
subalgebra or ideal
of L, return the matrix of (ad) xas an
element of a matrix Lie algebra.
> L:=LieAlgebra("B2",RationalField());
> AdjointMatrix(L, 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 0 0]
[ 0 0 0 0 0 0 0 0 0 0]
[ 1 0 0 0 0 0 0 0 0 0]
[ 2 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 1 0 0 0 0 0 0 0]
[ 0 0 0 0 0 -1 0 0 0 0]
a[i] : AlgMatLieElt, RngIntElt -> ModTupRngElt
If a is an element of a structure constant Lie algebra L of dimension n
and 1 ≤i≤n is a positive integer, then the i-th component
of the element a is returned (as an element of the base ring R of L).
If a is an element of a matrix Lie algebra L of degree n and
1 ≤i ≤n then the ith row of the matrix a is returned.
a[i] := r : AlgMatLieElt, RngIntElt, ModTupRngElt -> AlgMatLieElt
Given an element a belonging to a structure constant Lie algebra of dimension n
over R, a positive integer 1 ≤i≤n and an element r ∈R,
the i-th component of the element a is redefined to be r.
If a is an element of a matrix Lie algebra L of degree n over R and
1 ≤i ≤n, the ith row of the matrix a is redefined to be the
vector r over R.
a[i, j] := r : AlgMatLieElt, RngIntElt, RngIntElt, RngElt -> AlgMatLieElt
For an element a of a matrix Lie algebra L of degree n and integers
1 ≤i, j ≤n
return the element in the ith row and jth column of a or set this
element to be r where r is an element of the coefficient ring of L.
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