Journal of the
Korean Mathematical Society JKMS

The purpose of this paper is to study pseudo-Riemannian algebras, which are algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We find that pseudo-Riemannian algebras whose left centers are isotropic play a curial role and show that the decomposition of pseudo-Riemannian algebras whose left centers are isotropic into indecomposable non-degenerate ideals is unique up to a special automorphism. Furthermore, if the left center equals the center, the orthogonal decomposition of any pseudo-Riemannian algebra into indecomposable non-degenerate ideals is unique up to an isometry.

Keywords: pseudo-Riemannian algebra, indecomposable ideal, isometry, orthogonal decomposition

MSC numbers: Primary 17A30, 17D99