# Related Structures

In this section functions for creating other structures from a root system are briefly listed. The reader is referred to the appropriate chapters of the Handbook for more details.

## `RootDatum(R): RootSys -> RootDtm`

The (split) root datum corresponding to the root system $R$. The coefficients of the simple roots and coroots must be integral; otherwise an error is signalled. See Chapter [Root Data](../RootData/index-root-data.md#chaprootdtm)

## `CoxeterGroup(grpcat, R): Cat, RootSys -> grpcat`

The Coxeter group (of type `grpcat`) of a root system $R$. There are variations of this signature. The first argument can be `GrpMat`, `GrpPermCox`, `GrpPerm`, `GrpFPCox` or `GrpFP` and the second argument can be a root system or root datum. See Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox). If the first argument is `GrpFPCox` the braid group and pure braid group can be computed from the Coxeter group using the commands in Section [Braid Groups](../CoxeterGroups/group-braid.md#sectgrpcoxbraid).

## `CoxeterGroup(R): RootSys -> GrpPermCox`

## `WeylGroup(R): RootSys -> GrpPermCox`

The permutation Coxeter group with root system $R$. See Chapter [Coxeter Groups](../CoxeterGroups/index-coxeter-groups.md#chapgrpcox).

## `CoxeterGroup(GrpPermCox, R): Cat, RootSys -> RngIntElt`

## `ReflectionGroup(R): RootSys -> GrpMat`

The reflection group of the root system $R$. See Chapter [Reflection Groups](../ReflectionGroups/index-reflection-groups.md#chapgrprfl).

## `CoxeterGroup(GrpPermCox, W): Cat, RootSys -> GrpPermCox`

## `LieAlgebra(R, k): RootSys, Rng -> AlgLie`

The Lie algebra of the root system $R$ over the base ring $k$. See Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie).

## `MatrixLieAlgebra(R, k): RootSys -> GrpMat`

The matrix Lie algebra of the root system $R$ over the base ring $k$. See Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie).

## `Example: Related (ex-3411ae)`

```magma
> R := RootSystem("b3");
> SemisimpleType(LieAlgebra(R, Rationals()));
B3
> #CoxeterGroup(R);
48

```