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Properties of root data





IsIrreducible( RD ) : RootDtm -> BoolElt



True if the root datum RD is irreducible.



IsCrystallographic( RD ) : RootDtm -> BoolElt



True if the root datum RD is crystallographic.



IsSimplyLaced( RD ) : RootDtm-> BoolElt



True if the crystallographic root datum RD is simply laced (ie. its Dynkin diagram
contains no multiple bonds).



IsSemisimple( RD ) : RootDtm-> BoolElt



True if the root datum RD is semisimple (ie. its rank
is equal to its dimension).



IsAdjoint( RD ) : RootDtm-> BoolElt



True if the root datum RD is adjoint (ie. the
isogeny group is trivial).  RD must be crystallographic.



IsSimplyConnected( RD ) : RootDtm-> BoolElt



True if the root datum RD is simply connected
(ie. the isogeny group is equal to the fundamental group, ie. the
coisogeny group is trivial).  RD must be crystallographic.





Example RootDtm_Properties (H35E8)






> RD := RootDatum( "A5 B2" : Isogeny := "sc" );
> IsIrreducible( RD );
false
> IsCrystallographic( RD );
true
> IsSimplyLaced( RD );
false
> IsSemisimple( RD );
true
> IsAdjoint( RD );
false


For some of the exceptional Cartan types, there is only one isomorphism 
class of root data, which is both adjoint and simply connected.





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





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