Gradings induced by nilpotent elements

• Published in: Linear algebra and its applications Vol. 656; pp. 92 - 111
• Main Authors: García, Esther, Gómez Lozano, Miguel, Muñoz Alcázar, Rubén, Vera de Salas, Guillermo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2023
• Subjects: [formula omitted]-triple; Grading; Nilpotent element; von Neumann regular; 16W50; Nilpotent element; 16S50; 16E50; von Neumann regular; sl2-triple; Grading; 16W10
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2022.09.017

An element a is nilpotent last-regular if it is nilpotent and its last nonzero power is von Neumann regular. In this paper we show that any nilpotent last-regular element a in an associative algebra R over a ring of scalars Φ gives rise to a complete system of orthogonal idempotents that induces a finite Z-grading on R; we also show that such element gives rise to an sl2-triple in R with semisimple adjoint map adh, and that the grading of R with respect to the complete system of orthogonal idempotents is a refinement of the Φ-grading induced by the eigenspaces of adh. These results can be adapted to nilpotent elements a with all their powers von Neumann regular, in which case the element a can be completed to an sl2-triple and a is homogeneous of degree 2 both in the Z-grading of R and in the Φ-grading given by the eigenspaces of adh. Conversely, when there are enough invertible elements in Φ, we will show that if a nilpotent element a can be completed to an sl2-triple, then all powers ak are von Neumann regular.