In this section we describe the functions for working with the Hopf algebra structure of a quantized universal enveloping algebra (cf. Section Representations of Uq(L)).
The Hopf algebra structure that is used by default is the one described in Section Representations of Uq(L). As explained in that same section, it is possible to twist this by an automorphism, or an antiautomorphism.Given a quantized universal enveloping algebra U and (anti-) automorphisms f and g of U where g is the inverse of f (this is not checked by Magma) set U to use the corresponding twisted Hopf algebra structure.
This command has to be given before using the Hopf algebra structure, otherwise the default structure will be used. This includes creating a tensor product.
For some (anti-) automorphisms we refer to Section Automorphisms.
This function checks whether the quantized enveloping algebra U has been set to use a twisted Hopf structure. If the first value returned by this function is true, then the (anti-) automorphism and its inverse are also returned.
Returns the counit of the quantized enveloping algebra U. It is a map from U into the ground field of U.
Returns the antipode of the quantized enveloping algebra U. It is an antiautomorphism of U.
Returns the comultiplication of degree d of the quantized enveloping algebra U. This is a map from U into the d-fold tensor power of U. The comultiplication given in Section Representations of Uq(L) is of degree 2. The comultiplications of higher degree are obtained by repeating this map. So in particular, d has to be at least 2.An element of the d-fold tensor power of U is represented (rather primitively) by a list of d-tuples, each followed by a coefficient. The d-tuples are d-tuples of basis elements of U. The element represented by this list is the sum of the elements obtained by multiplying the i-th coefficient and the tensor product of the elements in the i-th d-tuple.
> U:= QuantizedUEA(RootDatum("A3"));
> d:= Comultiplication(U, 2);
> d(U.1);
[*
<1, F_1>,
1,
<F_1, K_1>,
1,
<F_1, [ K_1 ; 1 ]>,
(-q^2 + 1)/q
*]