# 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 $a$.