Let $A$ be an $n\times n$ generalized Cartan matrix. If $n-r$ is the rank of $A$ , then let $\fr{h}$ be a $n+r$ dimensional complex vector space. Choose $n$ linearly independent elements $\alpha_1,\ldots,\alpha_n\in\fr{h}^*$ (called roots), and $\check{\alpha}_1,\ldots,\check{\alpha}_n\in\fr{h}$ (called coroots) such that $\inn{\alpha_i,\check{\alpha_j}}=a_{ij}$ , where $\inn{\cdot,\cdot}$ is the natural pairing of $\fr{h}^*$ and $\fr{h}$ . This choice is unique up to automorphisms of $\fr{h}$ .

Then the Kac-Moody algebra associated to $\fr{g}(A)$ is the Lie algebra generated by elements $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ and the elements of $\fr{h}$ , with the relations

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.