# Construction of Elements

## `Zero(L): AlgLie -> AlgLieElt`

## `Zero(L): AlgMatLie -> AlgMatLieElt`

## `L ! 0: AlgLie, RngIntElt -> AlgLieElt`

## `L ! 0: AlgMatLie, RngIntElt -> AlgMatLieElt`

The zero element of the Lie algebra $L$.

## `Random(L): AlgLie -> AlgLieElt`

## `Random(L): AlgMatLie -> AlgMatLieElt`

Given a Lie algebra $L$ defined over a finite ring, a random element is returned.

## Construction of Elements of Structure Constant Algebras

### `elt<L | r₁, r₂, ..., rₙ>: AlgLie, RngElt, RngElt, ..., RngElt -> AlgLieElt`

Given a Lie algebra $L$ of dimension $n$ over a ring $R$, and ring elements $r_1, r_2, \ldots, r_n \in R$ construct the element $r_1 * e_1 + r_2 * e_2 + \ldots + r_n * e_n$ of $L$.

### `L ! Q: AlgLie, SeqEnum[RngElt] -> AlgLieElt`

Given a Lie algebra $L$ of dimension $n$ and a sequence $Q = [r_1, r_2, \ldots, r_n]$ of elements of the base ring $R$ of $L$, the element $r_1 * e_1 + r_2 * e_2 + \ldots + r_n * e_n$ of $L$ is constructed.

### `BasisProduct(L, i, j): AlgLie, RngIntElt, RngIntElt -> AlgLieElt`

Returns the product of the $i$-th and $j$-th basis element of the Lie algebra $L$.

### `BasisProducts(L): AlgLie -> SeqEnum`

```magma
Rep: MonStgElt                    Default: "Dense"
```

Returns the products of all basis elements of the Lie algebra $L$.

The optional parameter `Rep` may be used to specify the format of the result. If `Rep` is set to “Dense”, the products are returned as a sequence $Q$ of $n$ sequences of $n$ elements of $L$, where $n$ is the dimension of $L$. The element $Q[i][j]$ is the product of the $i$-th and $j$-th basis elements.

If `Rep` is set to “Sparse”, the products are returned as a sequence $Q$ containing quadruples $(i,j,k,a_{ijk})$ signifying that the product of the $i$-th and $j$-th basis elements is $\sum_{k=1}^n a_{ijk} b_k$, where $b_k$ is the $k$-th basis element and $n =$ dim($L$).

## Construction of Matrix Elements

Matrix Lie elements can be constructed using the functions below. For more information on constructing matrices see Section [Construction of a Matrix](../../Algebras/MatrixAlgebras/creation.md#algmat-elt-constr).

### `elt<R | L>: AlgMatLie, [ RngElt ] -> AlgMatLieElt`

### `elt<R | L>: AlgMatLie, RngElt, ..., RngElt -> AlgMatLieElt`

### `R ! L: AlgMatLie, [RngElt] -> AlgMatLieElt`

Create the element of the matrix Lie algebra $R$ of degree $n$ whose entries are the $n^2$ elements of the sequence $L$.

### `DiagonalMatrix(L, Q): AlgMatLie, [RngElt] -> AlgMatLieElt`

Diagonal matrix in the matrix Lie algebra $L$, given by the sequence $Q$ of ring elements.

### `ScalarMatrix(L, r): AlgMatLie, RngElt -> AlgMatLieElt`

Scalar matrix in the matrix Lie algebra $L$, defined by the ring element $r$.