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, ... , 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, ... , K_r; together with their inverses they generate the algebra U0. The final n generators are printed as E_1, ... , E_n. They generate a PBW-type basis of U^ +.
In U we use a basis of the integral form of U (Section The Z-form of Uq(L)). This means that instead of Fks and Eks we use the divided powers Fk(s) and Ek(s). Furthermore, a general basis element of U0 is a product of elements which are of the form [ Ki ; t ], or Ki[ Ki ; t ]. Here [ Ki ; t ] represents the "binomial" Ki choose t as described in Section The Z-form of Uq(L).
w0: SeqEnum Default: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.
> 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.
> 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
Assign the names in the sequence S to the generators of the algebra U.
Return the algebra identical to the algebra U but having coefficient ring R.