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## Lie Group behind a affine Lie algebra [closed]

Possible Duplicate:
Constructing Affine Kac-Moody Groups

Dear community,

I have the following question about affine-type Lie algebras.

In the finite-type Lie algebra, we associate to a root system a Lie algebra (or its enveloping algebra) that corresponds to a Lie group. For example, if we start from $A_n$ type, we have $sl(n)$, which is the Lie algebra of the group $SL(n)$.

Now, I start from an affine Cartan matrix $C$ and I build an affine-type Lie algebra. Like above, I can build $\widehat{sl}(n)$. Is it the Lie algebra of a Lie group ? If yes, which one is it ? The infinite dimension of the Lie algebra makes things unclear to me...

I accept both general answer and examples like in the $sl(n)$ case or even the $sl(2)$ case.