Why is the dimension of the cartan subalgebra $2n-\text{rank}(A)$ in the defintion from Kumar's book. From a few examples I can see why the defintion is the way it is, but, I would like a better understanding of why the dimension of the Cartan subalgebra is $2n-\text{rank}(A)$ and not $n$ or something else. I 


To be specific why does the Lie algebra associated to the universal extention $\tilde{L}SU(2)$ not have an associated Cartan matrix while the affine Lie algebra associated to $\mathbb{T}\ltimes\tilde{L}SU(2)$ does?



Kac Moody algebra defintion
I'll just elaborate on my comment from last year.


A Kac-Moody algebra is defined by generators-and-relations, with starting data given by an $n \times n$ matrix $A$. It can be written as $L \rtimes D$, where $D$ is a $n - rank(A)$-dimensional commutative Lie algebra that acts on $L$ by derivations. The standard examples are affine Lie algebras, where $D$ is one dimensional. It is common to throw away $D$ when studying these objects, because it doesn't really do much for the representation theory - the only information it adds is an "energy" grading. However, $D$ can help with some geometric intuition, because the Cartan subalgebra gets a non-singular inner product. This makes the geometry of reflections from the Weyl group more clear.


The standard example is with the loop algebra of $\mathfrak{sl}\_2$: if you forget about $D$, you lose the energy grading on the roots, and it is hard to draw a picture of them where they don't land on top of each other. When you add $D$, the weight space grows from a 2-dimensional space with singular inner product to the Lorentz space $\mathbb{R}^{2,1}$, the new "energy" axis is a line in the light cone (i.e., all vectors have norm zero), and the roots lie in a plane tangent to the light cone containing this axis. The Weyl group now acts by visible reflections, and the parabolic shape that shows up in characters of highest-weight representations can be seen as the intersection of a hyperboloid of constant norm with a plane parallel to the root space.


