# Homomorphisms to the Universal Enveloping Algebra

## `QUAToIntegralUEAMap(U): AlgQUE -> Map`

Given a quantized enveloping algebra $U$ returns the map from $U$ onto the integral form of the universal enveloping algebra of the corresponding Lie algebra (cf. Section [The ${\mathbb{Z}}$-form of $U_q(L)$](background.md#zform)). We refer to Section [Universal Enveloping Algebras](../LieAlgebras/universal_enveloping_algebras.md#sectunivea) for an account of universal enveloping algebras in Magma.

## `Example: QE Ato UEA (ex-08caa9)`

```magma
> U:= QuantizedUEA(RootDatum("C3"));
> f:= QUAToIntegralUEAMap(U);
> p:= CanonicalElements(U, [1,2,1]);
> [ f(u) : u in p ];
[
    y_1*y_2^(2)*y_3,
    2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5,
    y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_1*y_7,
    y_1*y_2^(2)*y_3 + y_2*y_3*y_4 - y_2*y_6,
    2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_2*y_3*y_4 - y_2*y_6 + y_4*y_5,
    2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + 2*y_2*y_3*y_4 - y_2*y_6 + y_4*y_5,
    y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_2*y_3*y_4 + y_1*y_7 + y_4*y_5 + y_8
]

```

So this allows one to construct elements of the canonical basis of a universal enveloping algebra (of a semisimple Lie algebra).