For a quantized enveloping algebra U this returns the bar-automorphism of U (Section Quantized Enveloping Algebras). The map returned by this function has its inverse stored, which can be retrieved using Inverse.
For a quantized enveloping algebra U this returns the automorphism of U that is denoted by ω (Section Quantized Enveloping Algebras). The map returned by this function has its inverse stored, which can be retrieved using Inverse.
For a quantized enveloping algebra U this returns the anti-automorphism of U that is denoted by τ (Section Quantized Enveloping Algebras). The map returned by this function has its inverse stored, which can be retrieved using Inverse.
Let U be a quantized enveloping algebra, and let k be an integer between 1 and the rank of the root datum. Then this function returns the automorphism Tαk of U, corresponding to the k-th simple root (Section PBW-type Bases). The map returned by this function has its inverse stored, which can be retrieved using Inverse.
Let U be a quantized enveloping algebra, and let p be a permutation of { 1, ..., r}, where r is the rank of the root datum. Here p must represent a diagram automorphism of the root datum (i.e., it leaves the Dynkin diagram invariant). Then this function returns the corresponding automorphism of U (see Section Quantized Enveloping Algebras). The map returned by this function has its inverse stored, which can be retrieved using Inverse.
> R:= RootDatum("G2");
> U:= QuantizedUEA(R);
> b:= BarAutomorphism(U);
> b(U.3);
(q^10 - q^6 - q^4 + 1)/q^4*F_1^(2)*F_6 + (q^4 - 1)/q^2*F_1*F_5 + F_3
A known result states that Tαr - 1 = τ Tαr
τ. We check that for the quantum group of type C3, and
the third simple root.
> U:= QuantizedUEA(RootDatum("C3"));
> t:= AntiAutomorphismTau(U);
> T:= AutomorphismTalpha(U, 3);
> Ti:= Inverse(T);
> f:= t*T*t;
> &and[ Ti(U.i) eq f(U.i) : i in [1..21] ];
true
A diagram automorphism maps the canonical basis into itself. We check
that for the set of elements of the canonical basis of the quantized
enveloping algebra of type D4 of weight
α1 + 3α2 + 2α3 + 2α4. (Here αi is the i-th
simple root.) The chosen diagram automorphism maps this weight to
2α1 + 3α2 + α3 + 2α4. Therefore we also compute
the elements of the canonical basis of that weight.
> U:= QuantizedUEA(RootDatum("D4"));
> p:= SymmetricGroup(4)!(1,3,4);
> d:= DiagramAutomorphism(U, p);
> e1:= CanonicalElements(U, [1,3,2,2]);
> e2:= CanonicalElements(U, [2,3,1,2]);
> &and[ d(x) in e2 : x in e1 ];
true