# extended Cartan matrix

Let $A$ be the Cartan matrix of a complex, semi-simple, finite dimensional, Lie algebra $\mathfrak{g}$. Recall that $A=(a_{ij})$ where $a_{ij}=\langle\alpha_{i},\alpha_{j}^{\vee}\rangle$ where the $\alpha_{i}$ are simple roots for $\mathfrak{g}$ and the $\alpha_{j}^{\vee}$ are simple coroots. The extended Cartan matrix denoted $\hat{A}$ 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 $\left\{\alpha_{1},\ldots,\alpha_{n}\right\}$) root for $\mathfrak{g}$. $\theta$ can be defined as a root of $\mathfrak{g}$ 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 $\hat{A}$ is an example of a generalized Cartan matrix. The corresponding Kac-Moody Lie algerba is said to be of affine type.

For example if $\mathfrak{g}=\mathfrak{sl}_{n}\mathbb{C}$ then $\hat{A}$ is obtained from $A$ by adding a zero-th row: $(2,-1,0,\ldots,0,-1)$ and zero-th column $(2,-1,0,\ldots,0,-1)$ simultaneously to the Cartan matrix for $\mathfrak{sl}_{n}\mathbb{C}$.

## References

• 1 Victor Kac, Infinite Dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990.
Title extended Cartan matrix ExtendedCartanMatrix 2013-03-22 15:30:14 2013-03-22 15:30:14 benjaminfjones (879) benjaminfjones (879) 8 benjaminfjones (879) Definition msc 17B67 GeneralizedCartanMatrix extended Cartan matrix