generalized Cartan matrix. Thus $2I - C$ is the adjacency matrix of an unoriented graph $\Gamma$ with $n$ vertices and no loop. Without loss of generality, we can assume that this graph is connected. We will often denote by $I = [1,n]$ the indexing set of rows and columns of $C$. Let $\h$ be a $\C$-vector space of dimension $2n-\rk(C)$. We choose linear independent subsets $\{h_i\mid i\in I\} \subset \h$ and $\{\a_i\mid i\in I\} \subset \h^*$ such that $\a_i(h_j) = c_{ij}$. Let $\g$ be the Kac-Moody Lie algebra over $\C$ with generators $e_i, f_i\ (i\in I)$, $h\in\h$, subject to the following relations: $$\begin{array}{l}
[h,h']=0,\ [h,e_i]={\a_i}(h)e_i,\ [h_,f_i]=-{\a_i}(h) f_i,\qquad (h,h'\in\h),\\[2mm]
[e_i,f_j]=\delta_{ij}h_i,  \hskip 5.7cm (i,j\in I),\\[3mm]
\ad(e_i)^{1-{c_{ij}}}(e_j) = 
\ad(f_i)^{1-{c_{ij}}}(f_j) = 0,\ \hskip 2.1cm(i\not = j).
\end{array}$$ We denote by $\n_+$ ($\n_-$) the subalgebra generated by $e_i\ (i\in I)$ ($f_i\ (i\in I)$). For simplicity we shall often write $\n$ instead of $\n_+$.

Let $W$ be the subgroup of $\GL(\h^*)$ generated by the reflexions $$s_i(\a) = \a - \a(h_i)\a_i,\qquad (i\in I,\ \a\in\h^*).$$ This is a Coxeter group with length function $w \mapsto \ell(w)$. For $\a\in\h^*$ let $$\g_\a = \{x\in \g \mid [h,x] = \a(h)x,\ h\in\h\}.$$ We denote by $\De := \{\a\in\h^*\mid \g_\a \not = 0\}$ 