Transformations of Dynkin Diagrams
Before we continue constructing root systems, we want to stop and observe a couple things about transformations of Dynkin diagrams.
First off, I want to be clear about what kinds of transformations I mean. Given Dynkin diagrams 
and 
, I want to consider a mapping 
that sends every vertex of to a vertex of . Further, if 
and 
are vertices of joined by 
edges, then 
and 
should be joined by edges in as well, and the orientation of double and triple edges should be the same.
But remember that and , as vertices, really stand for vectors in some base of a root system, and the number of edges connecting them encodes their Cartan integers. If we slightly abuse notation and write and for these bases, then the mapping defines images of the vectors in , which is a basis of a vector space. Thus extends uniquely to a linear transformation from the vector space spanned by to that spanned by . And our assumption about the number of edges joining two vertices means that preserves the Cartan integers of the base .
Now, just like we saw when we showed that the Cartan matrix determines the root system up to isomorphism, we can extend to a map from the root system generated by to the root system generated by . That is, a transformation of Dynkin diagrams gives rise to a morphism of root systems.
Unfortunately, the converse doesn’t necessarily hold. Look back at our two-dimensional examples; specifically, consider the 
and 
root systems. Even though we haven’t really constructed the latter yet, we can still use what we see. There are linear maps taking the six roots in to either the six long roots or the six short roots in . These maps are all morphisms of root systems, but none of them can be given by transformations of Dynkin diagrams. Indeed, the image of any base for would contain either two long roots in or two short roots, but any base of would need to contain both a long and a short root.
However, not all is lost. If we have an isomorphism of root systems, then it must send a base to a base, and thus it can be seen as a transformation of the Dynkin diagrams. Indeed, an isomorphism of root systems gives rise to an isomorphism of Dynkin diagrams.
The other observation we want to make is that duality of root systems is easily expressed in terms of Dynkin diagrams: just reverse all the oriented edges! Indeed, we’ve already seen this in the case of 
and 
root systems. When we get to constructing and 
, we will see that they are self-dual, in keeping with the fact that reversing the directed edge in each case doesn’t really change the diagram.
March 5, 2010 -
Posted by John Armstrong |
Geometry, Root Systems
The Unapologetic Mathematician 
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