Simplicity of certain infinite-dimensional Lie algebras (Q1891721)

Let \(A=(a_{ij})\) be a symmetrizable complex-valued \(n\times n\) matrix and let \(\widetilde G(A)\) be the Lie algebra with \(3n\) generators \((e_i, f_i,h_i)\) and the defining relations \([e_i,f_i]= \delta_{ij}h_i\), \([h_i,e]= a_{ij}e_j\), \([h_if_i]= -a_{ij}f_i\), and \([h_ih_j]=0\). Here the authors give necessary and sufficient conditions for the simplicity of \(\widetilde G(A)/ \widetilde C\), where \(\widetilde C\) is the center of \(\widetilde G(A)\), thus extending a result given by \textit{V. G. Kac} [Bull. Am. Math. Soc., New Ser. 2, 311-314 (1980; Zbl 0427.17012)].