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Morphisms are currently only defined for split root data.
Let Ri=(Xi, Φi, Yi, Φistar) be a root datum for i=1, 2.
A morphism of root data φ:R1to R2 consists of a pair of Z-linear
maps φX:X1to X2 and ΦY:Y1to Y2 satisfying
- 1.
- φX(Φ1)⊆Φ2∪{0}; and
- 2.
- φY(αstar)=φX(α)star (with the convention that
0star=0).
A fractional morphism is similar, except that it consists of Q-linear
maps on the (co)root spaces X1⊗Q to X2⊗Q and
Y1⊗Q to Y2⊗Q.
The main examples of fractional morphisms are isogeny maps
(Section Isogeny of Split Reduced Root Data).
A dual morphism is similar, except that the maps are X1to Y2 and
Y1to X2.
This is clearly equivalent to a morphism from R1 to the dual of R2.
Finally we define a dual fractional morphism in the obvious way.
A (fractional) morphism φ:R1to R2 also stores a sign corresponding to
each simple root of R1. This has no effect on the action of φon roots
or coroots, but does effect the definition of the corresponding homomorphisms
of Lie algebras and groups of Lie type.
hom<R -> S | phiX, phiY> : RootDtm, RootDtm, Mtrx, Mtrx -> Map
Construct a (fractional) morphism of root data R to S with the given linear
maps or matrices of linear maps.
Construct a (fractional) morphism of root data R to S with the given sequence of root images.
The sequence Q must have length 2N and consist of elements in the range
[0, ..., 2M],
where N is the number of positive roots of R and M is the number of positive roots of S.
The domain R must be semisimple.
Morphism(R, S, phiX, phiY) : RootDtm, RootDtm, Mtrx, Mtrx -> Map
SimpleSigns: . Default: 1
Check: BoolElt Default: true
Construct a (fractional) morphism of root data R to S with the given sequence of root images.
The sequence Q must have length 2N and consist of elements in the range
[0, ..., 2M],
where N is the number of positive roots of R and M is the number of positive roots of S.
The domain R must be semisimple.
SimpleSigns is a sequence of signs corresponding to the simple roots, or
+-1 to indicate a constant sequence.
If Check is set to false, the function does not check that the maps send
(co)roots to (co)roots.
This function is the same as the constructor hom, except for these
optional parameters.
SimpleSigns: . Default: 1
Check: BoolElt Default: true
Construct a (fractional) morphism of root data R to S with the given sequence
of root images.
The sequence Q must have length 2N and consist of elements in the range
[0, ..., 2M],
where N is the number of positive roots of R and M is the number of positive roots of S.
The domain R must be semisimple.
SimpleSigns is a sequence of signs corresponding to the simple roots, or
+-1 to indicate a constant sequence.
If Check is set to false, the function does not check that the maps send
(co)roots to (co)roots.
This function is the same as the constructor hom, except for these
optional parameters.
DualMorphism(R, S, phiX, phiY) : RootDtm, RootDtm, Mtrx, Mtrx -> Map
Check: BoolElt Default: true
Construct a (fractional) dual morphism of root data R to S with the given linear maps or matrices of linear maps.
If Check is set to false, the function does not check that the maps send
(co)roots to (co)roots.
Check: BoolElt Default: true
Construct a (fractional) dual morphism of root data R to S with the given sequence
of root images.
The sequence Q must have length 2N and consist of elements in the range
[0, ..., 2M],
where N is the number of positive roots of R and M is the number of positive roots of S.
The domain R must be semisimple.
If Check is set to false, the function does not check that the maps send
(co)roots to (co)roots.
The indices of the root images of the (dual) (fractional) morphism φ.
The indices of the root images of the automorphism φ.
Returns true if the morphism φis an isogeny, ie,
φY is onto with finite kernel.
IdentityAutomorphism(R) : RootDtm -> Map
The identity morphism R to R.
We construct the fractional morphism from the standard root datum of type
A3 onto the adjoint root datum of type A3.
This will allow us to construct the algebraic projection
(GL)4to(PGL)4 in Section Algebraic Homomorphisms.
> RGL := StandardRootDatum( "A", 3 );
> RPGL := RootDatum( "A3" );
> A := VerticalJoin( SimpleRoots(RGL), Vector([Rationals()|1,1,1,1]) )^-1 *
> VerticalJoin( SimpleRoots(RPGL), Vector([Rationals()|0,0,0]) );
> B := VerticalJoin( SimpleCoroots(RGL), Vector([Rationals()|1,1,1,1]) )^-1 *
> VerticalJoin( SimpleCoroots(RPGL), Vector([Rationals()|0,0,0]) );
> phi := hom< RGL -> RPGL | A, B >;
> v := Coroot(RGL,1);
> v; phi(v);
( 1 -1 0 0)
( 2 -1 0 )
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