# Related Structures

In this section we briefly list functions for creating other structures from a reflection group. See the appropriate chapters of the Handbook for more details.

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

The Coxeter group isomorphic to the real reflection group $W$. See Chapter ChapGrpPermCox.

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

The permutation Coxeter group isomorphic to the finite real reflection group $W$. See Chapter ChapGrpPermCox.

## `LieAlgebra(W, R): GrpMat, Rng -> AlgLie`

The reductive Lie algebra over the ring $R$ with Weyl group $W$. Unless $W$ is finite, real, and crystallographic, an error is flagged. See Section [Almost Reductive Lie Algebras](../LieAlgebras/family-construct.md#subsectalglieconstrred).

## `GroupOfLieType(W, k): GrpMat, Rng -> GrpLie`

The group of Lie type over the field $k$ with Weyl group $W$. Unless $W$ is finite, real, and crystallographic, an error is flagged. The roots and coroots of $W$ must also have integral components. See Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie).