Gradings induced by nilpotent elements (Q2093503)

In this paper the authors prove that any nilpotent last-regular element \(a\) in an associative algebra \(R\) over a ring of scalars \(\Phi\) gives rise to a complete system of orthogonal idempotents that induces a finite \(\mathbb{Z}\)-grading on \(R\); Further, they show that such element gives rise to an \(\mathfrak{sl_2}\)-triple in \(R\) with semisimple adjoint map \(ad_h\), and that the grading of \(R\) with respect to the complete system of orthogonal idempotents is a refinement of the \(\Phi\)-grading induced by the eigenspaces of \(ad_h\).