# Construction

## `QuantizedUEA(R): RootDtm -> AlgQUE`

## `QuantizedUEAlgebra(R): RootDtm -> AlgQUE`

## `QuantizedUniversalEnvelopingAlgebra(R): RootDtm -> AlgQUE`

```magma
w0: SeqEnum                    Default: 
```

This creates the quantized enveloping algebra $U$ corresponding to the root datum $R$. The algebra $U$ will be defined over the rational function field in one variable, $q$, over the rational numbers.

Let $n$ and $r$ respectively be the number of positive roots, and the rank of $R$. Then $U$ has $2n+r$ generators, accessible as `U.1`, `U.2` and so on. The first $n$ of these are printed as `F_1`, $\ldots$ , `F_n`. They generate a PBW-type basis of the subalgebra $U^-$ (cf. Section [PBW-type Bases](background.md#pbwbases)). The next $r$ generators are printed as `K_1`, $\ldots$ , `K_r`; together with their inverses they generate the algebra $U^0$. The final $n$ generators are printed as `E_1`, $\ldots$ , `E_n`. They generate a PBW-type basis of $U^+$.

In $U$ we use a basis of the integral form of $U$ (Section [The ${\mathbb{Z}}$-form of $U_q(L)$](background.md#zform)). This means that instead of $F_k^s$ and $E_k^s$ we use the divided powers $F_k^{(s)}$ and $E_k^{(s)}$. Furthermore, a general basis element of $U^0$ is a product of elements which are of the form $[ K_i ; t ]$, or $K_i[ K_i ; t ]$. Here $[ K_i ; t ]$ represents the “binomial” $K_i$ choose $t$ as described in Section [The ${\mathbb{Z}}$-form of $U_q(L)$](background.md#zform).

It is also possible to give a reduced expression for the longest element in the Weyl group, by setting the optional parameter `w0` equal to a sequence of indices lying between $1$ and the rank of $R$. If we replace each index by the corresponding simple reflection, then a reduced expression for the longest element in the Weyl group has to be obtained. In that case the PBW-basis relative to that sequence will be created (and used in subsequent computations). If this parameter is not given, then the lexicographically smallest reduced expression will be used.

## `Example: Q Grp Constr (ex-578e8e)`

We construct the quantum group corresponding to the root datum of type $C_3$.

```magma
> R:= RootDatum("C3");
> U:= QuantizedUEA(R);
> U.9; U.10; U.15;
F_9
K_1
E_3
> U.21*U.14*U.10*U.9*U.1;
1/q*F_1*F_9*K_1*E_2*E_9 - 1/q*F_1*F_9*K_1*E_6 + 1/q^3*F_1*K_1*[ K_3 ; 1 ]*E_2 -
    F_9*E_3*E_9 + F_9*E_8 - 1/q^2*[ K_3 ; 1 ]*E_3

```

Now we construct the same algebra, but use the PBW-basis relative to a different reduced expression of the longest element in the Weyl group.

```magma
> U:= QuantizedUEA(R : w0:= [2,3,1,2,3,1,2,3,1]);
> U.21*U.14*U.10*U.9*U.1;
q^2*F_1*F_9*K_1*E_2*E_9 + (q^2 - 1)/q^3*F_9*K_1*[ K_2 ; 1 ]*E_6*E_9 -
    1/q^2*F_9*K_1*K_2*E_6*E_9 - q^2*F_1*F_9*K_1*E_4 + q^2*F_1*K_1[ K_1 ; 1 ]*E_2 +
    q*F_3*K_1*E_2*E_9 + (-q^2 + 1)/q^3*F_9*K_1*[ K_2 ; 1 ]*E_7 +
    1/q^2*F_9*K_1*K_2*E_7 + (q^2 - 1)/q*K_1[ K_1 ; 1 ]*[ K_2 ; 1 ]*E_6
    - K_1[ K_1 ; 1 ]*K_2*E_6 - q*F_3*K_1*E_4 + q*F_1*E_2

```

## `AssignNames(U, S): AlgPBW, [ MonStgElt ]`

Assign the names in the sequence $S$ to the generators of the algebra $U$.

## `ChangeRing(U, R): AlgQUE, Rng -> AlgQUE`

Return the algebra identical to the algebra $U$ but having coefficient ring $R$.