Simplicity of Certain Infinite-Dimensional Lie Algebras Original Research Article

Let A = (aij) be a symmetrizable complex-valued n × n matrix and let image(A) be the Lie algebra with 3n generators {ei, fi, hi} and the defining relations [ei, fi] = δijhi, [hi, ej] = aijej, [hifi] = −aijfj, and [hihj] = 0. The purpose of this paper is to give necessary and sufficient conditions for simplicity of image(A)/ image, where image is the center of image(A), thus extending a result given by Kac.