Related Structures#

CoefficientRing(U): AlgQUE -> Fld#: This returns the ring of coefficients of the quantized enveloping algebra \(U\).

RootDatum(U): AlgQUE -> RootDtm#: This returns the root datum corresponding to the quantized enveloping algebra \(U\).

PositiveRootsPerm(U): AlgQUE -> SeqEnum#: Given a quantized universal enveloping algebra \(U\) with root datum \(R\) returns a sequence consisting of the integers between \(1\) and the number of positive roots of \(R\). If the \(k\)-th element of this sequence is \(m\), then the generator \(F_k\) of \(U\) is of weight \(-\beta_m\), where \(\beta_m\) is the \(m\)-th positive root of \(R\) (as returned by PositiveRoots(R)). (For the definition of weight of an element of \(U\) see Section PBW-type Bases.) Furthermore, the generator \(E_k\) is of weight \(\beta_m\).

Example: Q Grp Rel Str (ex-3ae424)#

> R:= RootDatum("D4");
> U:= QuantizedUEA(R);
> CoefficientRing(U);
Univariate rational function field over Rational Field
Variables: q
> RootDatum(U);
Adjoint root datum of type D4
> PositiveRootsPerm(U);
[ 1, 5, 2, 8, 6, 3, 12, 11, 9, 10, 7, 4 ]

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So for instance this means that \(F_6\) is of weight \(-\beta_3\).