Operations on Elements

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): AlgLie, AlgLieElt -> BoolElt
IsCentral(L, M): AlgMatLie, AlgMatLieElt -> BoolElt

Given an element \(x\) of the Lie algebra \(L\), return true if \(x\) is central in \(L\).

NonNilpotentElement(L): AlgLie -> AlgLieElt

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], §2.7.

Example: Non-Nilpotent Element (ex-a38e8a)

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)

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AdjointMatrix(L, x): AlgLie, AlgLieElt -> AlgMatLieElt
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 \({\rm ad} x\) as an element of a matrix Lie algebra.

Example: Adjoint Matrix of an Element (ex-c4c36f)
> 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]

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Indexing

a[i]: AlgLieElt, RngIntElt -> RngElt
a[i]: AlgMatLieElt, RngIntElt -> ModTupRngElt

If \(a\) is an element of a structure constant Lie algebra \(L\) of dimension \(n\) and \(1 \leq i\leq n\) 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 \(1 \leq i \leq n\) then the \(i\)th row of the matrix \(a\) is returned.

a[i] := r: AlgLieElt, RngIntElt, RngElt -> AlgLieElt
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 \(1 \leq i\leq n\), and an element \(r \in R\), 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 \(1 \leq i \leq n\), the \(i\)th row of the matrix \(a\) is redefined to be the vector \(r\) over \(R\).

a[i, j]: AlgMatLieElt, RngIntElt, RngIntElt -> RngElt
a[i, j] := r: AlgMatLieElt, RngIntElt, RngIntElt, RngElt -> AlgMatLieElt

For an element \(a\) of a matrix Lie algebra \(L\) of degree \(n\) and integers \(1 \leq i, j \leq n\) return the element in the \(i\)th row and \(j\)th column of \(a\) or set this element to be \(r\) where \(r\) is an element of the coefficient ring of \(L\).