Why are affine Lie algebras called affine?

Question

Hi. I was wondering if someone could explain why we call affine Lie algebras affine. Thanks!

Oliver

Answer 1

It's not easy to separate out the purely mathematical from the historical question here: What is the mathematical justification for use of the label "affine" and how did this label get attached to certain Lie algebras? The history in this case would be challenging to sort out, partly because some of the people involved are likely to remember it differently. Here is my own approach to answering both questions, which is not guaranteed to be exact.

The study of what are now usually called Kac-Moody algebras began in the mid-1960s with the simultaneous and independent thesis work by Kac (Moscow) and Moody (Toronto). Though their motivations differed, both of them arrived at a construction of (typically infinite dimensional) Lie algebras using generators and relations analogous to those found earlier in the study of finite dimensional simple Lie algebras over $\mathbb{C}$. The starting point is a generalized version of the classical Cartan matrix, leading to versions of root systems and Weyl groups as well.

But it took a while for terminology to settle down. For intance, Moody preferred at first the term "Euclidean Lie algebra" for what later became known as affine. This is probably related to a traditional geometric trichotomy: spherical, euclidean, hyperbolic. (Such terminology is not unreasonable in the study of generalized Cartan matrices: see for instance Cor. 15.11 in Carter's 2005 textbook Lie Algebras of Finite and Affine Type). However, by the 1970s the GCM's and the Lie algebras themselves were being classified as finite, affine, or indefinite type. Those of finite type are the classical ones, while the affine Lie algebras involve affine root systems and affine Weyl groups (as pointed out in comments here).

At first the work of Kac and Moody seemed to me to be a creative but standard sort of generalization popular in dissertations. But the striking 1972 paper by I.G. Macdonald on affine root systems and the Dedekind $\eta$-function, followed soon by Kac's explanation in terms of the "Weyl-Kac" character formula for an affine Lie algebra, gave the entire subject a much higher profile. (Connections with mathematical physics provided separate impetus, especially for the study of affine Lie algebras.)

To extract the term "affine" from all this history is not entirely straightforward. Certainly the affine Weyl groups were in use decades earlier in connection with compact Lie groups; these were also beginning to surface in the 1960s in the work of Iwahori-Matsumoto on BN-pairs and Bruhat decomposition in Chevalley groups over local fields. Generalizations of classical root systems were also in the air. But on balance it seems to me that the classification of generalized Cartan matrices using the above trichotomy provided the immediate rationale for the term "affine Lie algebra", with affine roots and affine Weyl groups not far behind.