# Structure of affine Lie algebras

It is well-known that to every simple Lie algebra $$\mathfrak{g}$$ one can associate an affine Kac-Moody algebra by a double extension (once by a 2-cocycle and once by a derivation). One can then show that this algebra is the Kac-Moody algebra associated to the extended Cartan matrix of $$\mathfrak{g}$$.

However it is the intermediate Lie algebra, obtained by the first central extension, that is of interest to me. This Lie algebra $$L\mathfrak{g}\oplus\mathbb{C}c$$ (where $$L\mathfrak{g}$$ is the loop algebra of $$\mathfrak{g}$$) with bracket $$[g\otimes t^k+\lambda c, g'\otimes t^l+\mu c] = [g\otimes t^k, g'\otimes t^l]_{L\mathfrak{g}} + k(g|g')\delta_{k+l, 0}$$is often called the ''affine Lie algebra'' associated to $$\mathfrak{g}$$.

Since in some papers the further extension by a derivation is omitted I was wondering what the structure of this ''affine Lie algebra'' is. Does this algebra admit a generalized Cartan matrix (is it a Kac-Moody algebra on its own)? And perhaps a secondary question, why do some authors (especially in the physics literature) omit this second extension?