I am reading Kac's Infinite dimensional Lie algebras, third edition. In section 7.12, Kac discusses completions of Kac-Moody algebras of infinite rank, and define $\bar{\mathfrak{g}}(A)$, for any infinite generalized Cartan matrix $A$ with only finitely many nonzero entries in every row and column, in such a way that the completed Cartan subalgebra $\bar{\mathfrak{h}}$ is $\prod_i \mathbb{C} \alpha_i^\vee$.
He then mentions that, for $A$ the Cartan matrix of type $A_\infty$, $\bar{\mathfrak{g}}(A)$ is clearly isomorphic to the algebra $\overline{gl}_\infty$ of infinite matrices $(a_{ij})_{i,j\in \mathbb{Z}}$ with $a_{ij}=0$ for $|i-j|$ large enough. That statement seems incorrect to me with the previous definition of $\bar{\mathfrak{g}}(A)$, as the Cartan subalgebras don't seem to match. Indeed, there is a surjection from $\prod_i \mathbb{C} \alpha_i^\vee$ to the diagonal matrices in $\overline{gl}_\infty$, which sends $\sum c_i \alpha_i^\vee$ to $\sum c_i \left(E_{i,i}-E_{i+1,i+1}\right)$, but this is not an isomorphism: the kernel is spanned by $\sum_i \alpha_i^\vee$. In fact I believe that $\bar{\mathfrak{g}}(A)$ should really be the central extension $a_\infty$ of $\overline{gl}_\infty$ which is described in the rest of the section. Is that right? Is that a mistake in Kac's book?
