Finite \(\mathbb{Z}\)-gradings of simple associative algebras. (Q852918)

The author proves that certain \(\mathbb{Z}\)-graded algebras with finite support have non-trivial 3-gradings (i.e. \(\mathbb{Z}\)-gradings with support contained in the set \(\{-1,0,1\}\)). As a consequence, every finite \(\mathbb{Z}\)-grading of a simple associative algebra \(A\) is shown to arise from a Peirce decomposition (in the sense of \textit{O. N. Smirnov} [J. Algebra 196, No. 1, 171-184 (1997; Zbl 0899.16023)]) induced by a complete system of orthogonal idempotents lying in the maximal left quotient algebra of \(A\). Also it is proved that such an \(A\) has a 3-grading. Some equivalent characterizations of left non-singularity are given for certain \(\mathbb{Z}\)-graded algebras. An application to finite \(\mathbb{Z}\)-gradings on simple Lie algebras is given.