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

If the matrix $A$ is positive-definite, we obtain a finite dimensional semi-simple Lie algebra, and $A$ is the Cartan matrix associated to a Dynkin diagram. Otherwise, the algebra we obtain is infinite dimensional and has an $r$ -dimensional center.