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I'm learning about affine Lie algebras partly from J. Fuchs' Affine Lie Algebras and Quantum Groups. Actually I'm using Kerf and Bauerle but for this question I find Fuchs' labelling of the Dynkin diagrams to be much more logical.

What is the intuitive difference between twisted and untwisted affine Lie Algebras? It's not obvious to me from the Dynkin diagrams - for example those for $C_r^{(1)}$ (untwisted) and $B_r^{(2)}$ (twisted) differ only by the reversal of their arrows. This would imply to me from reading Kerf and Bauerle that if $C_r^{(1)}$ is the algebra obtained from the generalised Cartan matrix $A$ then $B_r^{(2)}$ is that obtained from $A^T$. What is twisty about that?

Cr1 Br2

Fuchs (p98) talks a bit about the differences in terms of the ability to remove the zeroth node and recover the finite subalgebra with correct Coxeter labels (which I can imagine is different, since $A$ and $A^T$ have dual Coxeter labels), and the automorphism groups of the diagrams, but what does this mean? I would be grateful for a pointer to any reference which discusses it, particularly if there was an insight into which algebra types crop up where in physics.

quixot
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Okay, so I came across a statement in p38 of these notes:

More precisely, the twisted Lie algebras are obtained when instead of $f(e^{2 \pi i} z)=f(z)$ one imposes the twisted boundary conditions $x \otimes f(e^{2\pi i})= \omega(x) \otimes f(z)$ for all $x \in \bar{\mathfrak{g}}$, where $\omega$ is an automorphism of the horizontal subalgebra of finite order $N$. In other words, $f$ is now no longer a function on the circle $S^1$, but rather on an $N$-fold covering of $S^1$.

So the terminology at least seems to be inherited from the manifestation of Affine algebras as twice extended loop algebras, but it's still unclear to me why that would swap $A$ for $A^T$. I guess I'll find out!

quixot
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