I am confused by the way that mathematicians and physicists use the words "Kac Moody algebra", and "loop algebra", and how exactly these concepts relate to one another. I will write down what I understood from different books, and ask you to help me connect the dots.
Following [1], this is what I understood:
A Kac Moody algebra is a Lie any Lie algebra obtained using (a generalization of) the Serre's construction from a generalized Cartan matrix, so that a Kac Moody algebra is a generalization of semisimple Lie algebras.
There are three types of Kac Moody algebras: semisimple, affine and indefinite. The affine Kac Moody algebras are infinite dimensional and have a subset called untwisted affine algebras which are extremely important in physics. Each of these types of Kac Moody algebras can be defined by properties of the generalized Cartan matrices from which they are obtained.
An explicit way to obtain any untwisted affine algebra is by starting with a loop algebra $\bar g = g\otimes\mathbb{C}[z,z^{-1}]$ (the Lie algebra of a loop group of a compact connected simple Lie group $G$ which has Lie algebra $g$) which has commutation relations (on a certain basis)
\begin{equation}
\left[T_{m}^{a}, T_{n}^{b}\right]=i f^{a b c} T_{m+n}^{c}
\end{equation}
and then take its universal (one dimensional) central extension $\hat g = \bar g \oplus K\mathbb C$. This has commutation relations (choosing generators such that the structure constants are fully antisymmetric - which we can since $g$ is simple and compact):
\begin{align}
[T^a_m,T^b_n] &= f^{ab}_{\ \ c} T^c_{m+n} + m K \delta^{ab}\delta_{m,-n} \label{eq:LoopAlgebraComutators}\\
[T^a_m, K] &= 0
\end{align}
Finally, one still has to perform a so called an extension by a derivation in order to obtain an untwisted affine Kac Moody algebra. This last step seems to be ignored by physicists, and I did not explore it.
Following [2], some things seem quite different:
They call $\bar g$ the untwisted Kac Moody algebra (and use it as a synonym for loop algebra). They still construct $\hat g$ and also call it Kac Moody algebra, but only with the excuse that central extensions are relevant for quantization. They never refer to derivation extension at all.
In [3] they say explicitly that $\hat g$ is an extension of a Kac Moody algebra, called the current algebra. So again they seem to see $\bar g$ as the Kac Moody algebra.
Questions:
1. Does the loop algebra have a universal central extension which happens to be one-dimensional or does is this extension universal only within the class of one-dimensional central extensions?
2. It worries me that they say in [2] (and many other physics texts and papers) that $\bar g$ and $\hat g$ are untwisted affine algebras - because sometimes they may use results that mathematicians proved for Kac Moody algebras, when physicists are not really working with them. do somehow most results that hold for untwisted affine algebras also hold for $\hat g$?
3. I concluded that Kac Moody algebras are actually not that important in physics, but what is important is the central extension of loop algebras. Is this correct?
4. Is what I wrote in the paragraph following [1] correct?
References:
[1] GGA Bäuerle and EA De Kerf. Lie algebras, part 1: Finite and infinite
dimensional lie algebras and applications in physics, 1990.
[2] Ralph Blumenhagen, Dieter Lüst, and Stefan Theisen. Basic concepts
of string theory. Springer Science & Business Media, 2012.
[3] Ralph Blumenhagen and Erik Plauschinn. Introduction to conformal
field theory: with applications to string theory, volume 779. Springer
Science & Business Media, 2009.
