# Kac-Moody algebra

Let $A$ be an $n\times n$ generalized Cartan matrix. If $n-r$ is the rank of $A$, then let $\mathfrak{h}$ be a $n+r$ dimensional complex vector space. Choose $n$ linearly independent  elements $\alpha_{1},\ldots,\alpha_{n}\in\mathfrak{h}^{*}$ (called roots), and $\check{\alpha}_{1},\ldots,\check{\alpha}_{n}\in\mathfrak{h}$ (called coroots) such that $\langle\alpha_{i},\check{\alpha_{j}}\rangle=a_{ij}$, where $\langle\cdot,\cdot\rangle$ is the natural pairing of $\mathfrak{h}^{*}$ and $\mathfrak{h}$. This choice is unique up to automorphisms   of $\mathfrak{h}$.

Then the Kac-Moody algebra associated to $\mathfrak{g}(A)$ is the Lie algebra  generated by elements $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}$ and the elements of $\mathfrak{h}$, with the relations

 $\displaystyle[X_{i},Y_{i}]$ $\displaystyle=\check{\alpha_{i}}$ $\displaystyle[X_{i},Y_{j}]$ $\displaystyle=0$ $\displaystyle=\alpha_{i}(h)X_{i}$ $\displaystyle[Y_{i},h]$ $\displaystyle=-\alpha_{i}(h)Y_{i}$ $\displaystyle\underbrace{[X_{i},[X_{i},\cdots,[X_{i}}_{1-a_{ij}\text{ times}},% X_{j}]\cdots]]$ $\displaystyle=0$ $\displaystyle\underbrace{[Y_{i},[Y_{i},\cdots,[Y_{i}}_{1-a_{ij}\text{ times}},% Y_{j}]\cdots]]$ $\displaystyle=0$

for any $h\in\mathfrak{h}$.