extended Cartan matrix

Let $A$ be the Cartan matrix of a complex, semi-simple, finite dimensional, Lie algebra $𝔤$. Recall that $A=(a_{ij})$ where $a_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$ where the ${\alpha }_i$ are simple roots for $𝔤$ and the ${\alpha }_j^{\vee }$ are simple coroots.The extended Cartan matrix denoted $\widehat{A}$ is obtained from $A$ by adding a zero-th row and column corresponding to adding a new simple root ${\alpha }_0:=-\theta$ where $\theta$ is the maximal (relative to $\{{\alpha }_1,\mathrm{\dots },{\alpha }_n\}$) root for $𝔤$. $\theta$ can be defined as a root of $𝔤$ such that when written in terms of simple roots $\theta ={\sum }_i{m}_i{\alpha }_i$ the coefficient sum ${\sum }_i{m}_i$ is maximal (i.e. it has maximal height). Such a root can be shown to be unique.

The matrix $\widehat{A}$ is an example of a generalized Cartan matrix. The corresponding Kac-Moody Lie algerba is said to be of affine type.

For example if $𝔤=𝔰{𝔩}_nℂ$ then $\widehat{A}$ is obtained from$A$ by adding a zero-th row: $(2,-1,0,\mathrm{\dots },0,-1)$ and zero-th column $(2,-1,0,\mathrm{\dots },0,-1)$ simultaneously to the Cartan matrix for $𝔰{𝔩}_nℂ$.