Construction

QuantizedUEA(R): RootDtm -> AlgQUE

QuantizedUEAlgebra(R): RootDtm -> AlgQUE

QuantizedUniversalEnvelopingAlgebra(R): RootDtm -> AlgQUE

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). 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 \(U_q(L)\)). 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 \(U_q(L)\).

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 \(C_3\).

> 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

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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.

> 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

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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 \(R\).