Properties of Root Data

IsFinite( R ) : RootSys -> BoolElt

Returns true.

IsIrreducible( R ) : RootDtm -> BoolElt

Returns true if, and only if, the root datum R is irreducible.

IsSemisimple( R ) : RootDtm-> BoolElt

Returns true if, and only if, the root datum R is semisimple, i.e. its rank is equal to its dimension.

IsCrystallographic( R ) : RootSys -> BoolElt

Returns true.

IsSimplyLaced( R ) : RootDtm-> BoolElt

Returns true if, and only if, the root datum R is simply laced, i.e. its Dynkin diagram contains no multiple bonds.

IsAdjoint( R ) : RootDtm-> BoolElt

Returns true if, and only if, the root datum R is adjoint, i.e. its isogeny group is trivial.

IsSimplyConnected( R ) : RootDtm-> BoolElt

Returns true if, and only if, the root datum R is simply connected, i.e. its isogeny group is equal to the fundamental group, i.e. its coisogeny group is trivial.

Example RootDtm_Properties (H80E9)

> R := RootDatum( "A5 B2" : Isogeny := "SC" );
> IsIrreducible( R );
false
> IsSimplyLaced( R );
false
> IsSemisimple( R );
true
> IsAdjoint( R );
false

> R := RootDatum( "G2" );
> IsAdjoint( R );
true
> IsSimplyConnected( R );
true