# Kac Moody algebra defintion

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?

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Probably part of the problem is that "the" definition doesn't really exist, just a range of closely related definitions. The ideas come from the independent thesis work of Kac and of Moody, but later developments tweaked the notions in various ways. It's a good idea to say more explicitly what version you are looking at. –  Jim Humphreys Sep 5 at 21:21
I find this dimension in every definition I have read. The Serre relations are all implemented slightly differently, but the dimension of the Cartan subalgebra is always $n+\text{corank}(A)$. –  Sven Cattell Sep 5 at 21:44
One reason why the extra dimensions are useful is that you end up with a nondegenerate inner product on the weight space. In your first example, your weight space becomes a singular plane embedded in $\mathbb{R}^{2,1}$, and it's harder to see the geometry attached to reflections. –  S. Carnahan Sep 5 at 21:47
I think you have a good answer to my admittingly soft question. –  Sven Cattell Sep 6 at 14:04