Why are affine Lie algebras called affine?

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

Oliver

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I always assumed it was because they correspond to affine reflection groups in the way that finite-dimensional semisimple algebras correspond to spherical (Euclidean) reflection groups. – Noam D. Elkies May 19 '13 at 0:29
That's what I thought too. I wouldn't be surprised if 'affine Coxeter diagrams' or 'affine Dynkin diagrams' were well-known long before people seriously started studying the corresponding Lie algebras. – John Baez May 19 '13 at 1:22
Another possibility is that the roots of an affine Lie algebra can be identified with affine linear forms on the Lie algebra of a Cartan subalgebra of the corresponding finite dimensional semisimple Lie algebra. See e.g. pg 71-72 of Loop Groups by Pressley, Segal. – solbap May 19 '13 at 4:42
The roots are "affine linear functions" on the "affine linear Euclidean" space. Here is a thesis with emphasis on the "affine aspect":science.uva.nl/onderwijs/thesis/centraal/files/f91068273.pdf – Dietrich Burde May 19 '13 at 8:45

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.
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.)