The Unapologetic Mathematician

Mathematics for the interested outsider

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 X and Y, I want to consider a mapping \phi that sends every vertex of X to a vertex of Y. Further, if \xi_1 and \xi_2 are vertices of X joined by n edges, then \phi(\xi_1) and \phi(\xi_2) should be joined by n edges in Y as well, and the orientation of double and triple edges should be the same.

But remember that \xi_1 and \xi_2, 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 X and Y for these bases, then the mapping \phi defines images of the vectors in X, which is a basis of a vector space. Thus \phi extends uniquely to a linear transformation from the vector space spanned by X to that spanned by Y. And our assumption about the number of edges joining two vertices means that \phi preserves the Cartan integers of the base X.

Now, just like we saw when we showed that the Cartan matrix determines the root system up to isomorphism, we can extend \phi to a map from the root system generated by X to the root system generated by Y. 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 A_2 and G_2 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 A_2 to either the six long roots or the six short roots in G_2. 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 A_2 would contain either two long roots in G_2 or two short roots, but any base of G_2 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 B_n and C_n root systems. When we get to constructing G_2 and F_4, 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 | Geometry, Root Systems | 3 Comments

   

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