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I have understood quite well how we construct Dynkin diagrams. My question is the following:

What immediate information can I extract just by looking at a Dynkin diagram?

Of course I can understand if it is an A,B,C,D,G,E or F type of algebra and I can count the number of positive simple roots. But what else can I find? Is it possible to extract algebra(group) isomorphism as it is done with the simple $B_2=A_1 + A_1$ example? Other information?

Marion
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1 Answers1

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The answer depends on what we understand by "just by looking at a Dynkin diagram". I guess, just by looking in the strict sense only gives the type, the rank, the dimension, and if the diagram is connected or not, i.e., if the algebra is simple or only semisimple. Maybe we also see immediately automorphisms.
If we allow a second of further thought, then we also know the Cartan matrix and the Serre relations, e.g., the Lie brackets. Given more time, we can construct the whole root system, and all algebraic structure coming with it, i.e., everything in a sense.

Edit: If the Dynkin diagram has $n_{ij}$ edges, then the Cartan numbers $A_{ij}$ are determined by the Diophantine equation $n_{ij}=A_{ij}A_{ji}$ for $i\neq j$, $A_{ii}=2$ and the condition $A_{ij}=0,-1,-2,-3$. The Serre relations now give information on the Lie brackets.

Dietrich Burde
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  • Hi and thanks for your answer. Could you give an example (pick a small rank) and tell me how do you see from a Dynkin diagram the Cartan matrix and the Lie brackets of the corresponding algebra? – Marion Mar 31 '15 at 13:33
  • You are welcome. Try $A_1$ or $A_2$ for an example of small rank. – Dietrich Burde Mar 31 '15 at 17:38