The Natural Module

Module(L): AlgLie -> ModTupRng

The module \(R^n\) underlying the Lie algebra \(L\).

RModule(L): AlgMatLie -> ModTupRng

The module \(R^n\) acted on by the matrix Lie algebra \(L\).

BaseModule(L): AlgMatLie -> ModTupRng

The space \(R^n\) acted on by the matrix Lie algebra \(L\).

Degree(L): AlgLie -> RngIntElt

Degree(L): AlgMatLie -> RngIntElt

The degree of the Lie algebra \(L\). If \(L\) is a structure constant algebra, this is just the dimension of \(L\). If \(L\) is a matrix Lie algebra, this is the degree of the matrices in \(L\).

Degree(a): AlgLieElt -> RngIntElt

Given an element \(a\) belonging to the Lie algebra \(L\), the dimension of \(L\) is returned.

ElementToSequence(a): AlgLieElt -> SeqEnum

Eltseq(a): AlgLieElt -> SeqEnum

The sequence of coefficients of the Lie element \(a\).

Coordinates(M, a): AlgLie, AlgLieElt -> SeqEnum

Coordinates(M, a): AlgMatLie, AlgMatLieElt -> SeqEnum

Let \(a\) be an element of a Lie algebra \(L\) and let \(M\) be a subalgebra of \(L\) containing \(a\). This function returns the coefficients of \(a\) with respect to the basis of \(L\).

InnerProduct(a, b): AlgLieElt, AlgLieElt -> RngElt

The (Euclidean) inner product of the coefficient vectors of \(a\) and \(b\), where \(a\) and \(b\) are elements of some Lie algebra.

Support(a): AlgLieElt -> SetEnum

The support of the Lie algebra element \(a\); i.e. the set of indices of the non-zero components of \(a\).