Saturated sets for generalized Cartan matrices (Q1095240)

Let \(A=(a_{ij})\) be a generalized Cartan matrix, and (\({\mathfrak h},\Pi,\Pi^{\vee})\) be a realization of A, where \({\mathfrak h}\) is a C- vector space and \(\Pi^{\vee}=\{h_ 1,...,h_ n\}\), \(\Pi =\{\alpha_ 1,...,\alpha_ n\}\) are linearly independent subsets of \({\mathfrak h}\) and the dual space \({\mathfrak h}^*\) of \({\mathfrak h}\) respectively such that \(\alpha_ i(h_ j)=a_{ji}\) for all i,j. A partial order \(<_{\Pi}\) in \({\mathfrak h}^*\) is defined by \(\mu <_{\Pi} \nu\) if and only if \(\nu\)-\(\mu\in \sum {\mathbb{Z}}_{\geq 0}\alpha_ i\). A non-empty subset S of the set of integral weights is called saturated if for all \(\mu\in S\) and for all i (1\(\leq i\leq n)\), \(\mu -k\alpha_ i\in S\) for all k such that \(0\leq k\leq \mu (h_ i)\). Let \({\mathfrak g}\) be the Kac-Moody Lie algebra associated with A. For an integrable \({\mathfrak g}\)-module V, the set of weights of V is denoted by \(\Lambda\) (V), and for a \({\mathfrak g}\)-module \({\mathfrak g}\) with respect to the adjoint action, \(\Delta =\Lambda ({\mathfrak g})\) is called the root system of \({\mathfrak g}\) and the element of \(\Delta\) is called a root. The authors show that \(\Delta\) is characterized by the following properties: (1) \(\Delta\) is a saturated set, (2) \(\Delta =-\Delta\), (3) \(k\alpha_ i\in \Delta\) if and only if \(k=0\), \(\pm 1\) for all i (1\(\leq i\leq n)\) and \(k\in {\mathbb{Z}}\), (4) \(\beta <_{\Pi} 0\) or \(0<_{\Pi} \beta\) for each \(\beta\in \Delta\), (5) If \(\beta =\sum c_ i\alpha_ i\in \Delta\), \(ht(\beta)=\sum c_ i>1\), then there exists some \(\alpha_ i\in \Pi\) such that \(\beta -\alpha_ i\in \Delta\). For a dominant weight \(\lambda\), let \(\Lambda\) (\(\lambda)\) be the set of weights of the standard \({\mathfrak g}\)-module whose highest weight is \(\lambda\). The authors also characterize the generalized Cartan matrix A for which the set of the negative imaginary roots \(\Delta_-^{im}\) is equal to the set \(\Lambda\) (\(\lambda)\) for some dominant weight \(\lambda\).