Algebraic Homomorphisms

GroupOfLieTypeHomomorphism(phi, k): Map, Rng → .

The algebraic homomorphism of groups of Lie type over the ring \(k\) corresponding to the root datum morphism \(\phi\). See Chapter ChapGrpLie.

Example: Creating Root Data Homomorphisms

This example constructs the algebraic projection \({GL}_4({\mathbb{Q}})\to{PGL}_4({\mathbb{Q}})\).

> 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 := GroupOfLieTypeHomomorphism( hom< RGL -> RPGL | A, B >, Rationals() );
> GL := Domain( phi );
> phi( elt<GL|<1,2>, Vector([Rationals()| 7,1,11,1])> );
x1(2) (   7 1/11   11)