# Related Structures

## `CoefficientRing(U): AlgQUE -> Fld`

This returns the ring of coefficients of the quantized enveloping algebra $U$.

## `RootDatum(U): AlgQUE -> RootDtm`

This returns the root datum corresponding to the quantized enveloping algebra $U$.

## `PositiveRootsPerm(U): AlgQUE -> SeqEnum`

Given a quantized universal enveloping algebra $U$ with root datum $R$ returns a sequence consisting of the integers between $1$ and the number of positive roots of $R$. If the $k$-th element of this sequence is $m$, then the generator $F_k$ of $U$ is of weight $-\beta_m$, where $\beta_m$ is the $m$-th positive root of $R$ (as returned by `PositiveRoots(R)`). (For the definition of weight of an element of $U$ see Section [PBW-type Bases](background.md#pbwbases).) Furthermore, the generator $E_k$ is of weight $\beta_m$.

## `Example: Q Grp Rel Str (ex-3ae424)`

```magma
> R:= RootDatum("D4");
> U:= QuantizedUEA(R);
> CoefficientRing(U);
Univariate rational function field over Rational Field
Variables: q
> RootDatum(U);
Adjoint root datum of type D4
> PositiveRootsPerm(U);
[ 1, 5, 2, 8, 6, 3, 12, 11, 9, 10, 7, 4 ]

```

So for instance this means that $F_6$ is of weight $-\beta_3$.