Start off with your favorite lie algebra g. Now take functions from an S^1 to g. Thinking of the S^1 as the unit circle in the complex plane, this becomes the space 
. This is an infinite Lie algebra, so unlike finite dimensional Lie algebras which we can't do much with, we can mess with this one. We can put in an anomaly c. This makes the commutation relations 
instead of the boring one we had before. What is the Cartan for this system? It includes the Cartan for g as g*t^0 and c. But there is a problem with this, we have no dependence on the power of t. This makes all the eigenspaces of the Cartan acting with ad(X) infinitely dimensional because we can multiply by any power of t to get something else in the eigenspace. This is why we extend the algebra once more by an operator that reads of the power of t like t*d/dt. This gives the full Cartan of our untwisted affine Lie algebra as what we had before and this thing which reads off the power of t. So we got rid of the degeneracy and we are set. We specify a weight by saying what its eigenvalue is under each of the ad(X)'s in the Cartan. Of course we said c is central so ad(c) should have eigenvalue 0 for the weights of the adjoint representation called roots. So roots are of the form (l,0,n) where l is a root of g. This means they are part of an r+1 dimensional lattice. r from g and one more from n. We can pick the simple roots to be (b,0,0) with b a simple root of g and one extra (-t,0,1) where t is the highest root of g. Doing the same trick of Dynkin diagrams as with ordinary Lie algebras indicates we just need to add one more node that represents that last affine root.
. This is an infinite Lie algebra, so unlike finite dimensional Lie algebras which we can't do much with, we can mess with this one. We can put in an anomaly c. This makes the commutation relations 
instead of the boring one we had before. What is the Cartan for this system? It includes the Cartan for g as g*t^0 and c. But there is a problem with this, we have no dependence on the power of t. This makes all the eigenspaces of the Cartan acting with ad(X) infinitely dimensional because we can multiply by any power of t to get something else in the eigenspace. This is why we extend the algebra once more by an operator that reads of the power of t like t*d/dt. This gives the full Cartan of our untwisted affine Lie algebra as what we had before and this thing which reads off the power of t. So we got rid of the degeneracy and we are set. We specify a weight by saying what its eigenvalue is under each of the ad(X)'s in the Cartan. Of course we said c is central so ad(c) should have eigenvalue 0 for the weights of the adjoint representation called roots. So roots are of the form (l,0,n) where l is a root of g. This means they are part of an r+1 dimensional lattice. r from g and one more from n. We can pick the simple roots to be (b,0,0) with b a simple root of g and one extra (-t,0,1) where t is the highest root of g. Doing the same trick of Dynkin diagrams as with ordinary Lie algebras indicates we just need to add one more node that represents that last affine root.
The additional node is shown in green. As usual the connections indicate values of the Cartan matrix which is now just one dimension bigger. We could even do more if we twisted by not insisting on a genuine map from the circle to the algebra, but one that differed by some automorphism upon going around the circle. A Z_2 automorphism would be like choosing antiperiodic boundary conditions instead of periodic ones, but no one can tell the difference between + and - so we could have chosen either.
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