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.
> L:=LieAlgebra("G2",RationalField());
> NonNilpotentElement(L);
(0 0 0 0 0 1 0 0 0 0 0 0 0 0)
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]
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.
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.
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.