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QuantizedUEAlgebra(R) : RootDtm -> AlgQUE
QuantizedUniversalEnvelopingAlgebra(R) : RootDtm -> AlgQUE
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.
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
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.
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