Simple associative algebras with finite \(\mathbb{Z}\)-grading
All algebras considered are linear (associative or Lie) algebras over a fixed commutative ring. A Peirce system in an algebra \(R\) is a system of submodules \(R_{p,q}\) (\(0\leq p,q\leq n\)) such that \(R_{p,q}R_{s,t}=\delta_{q,s}R_{p,t}\), \(\sum R_{p,q}=R\). If the latter sum is direct, the system is called strict. For example if \(e_0,e_1,\dots,e_n\) is a complete set of orthogonal idempotents in a ring \(R\), then the \(e_pRe_q\) form a strict Peirce system. Any Peirce system \(\{R_{p,q}\}\) induces a pregrading \(R=\sum R_k\) where \(R_k\) is the sum of all \(R_{p,p-k}\), which is a grading for a strict Peirce system. The author uses this setup to study simple graded algebras; for such an algebra \(R=\bigoplus R_k\) with lowest term \(R_{-n}\) he finds that the \(H_p=R_pR_{-n}R_{n-p}\) form a system of submodules which is orthogonal: \(H_iH_j=0\) (\(i<j\)) and complete: \(HRH=R\), where \(H=\sum H_i\), and is maximal subject to these conditions. Secondly he considers graded Lie algebras and uses the classification of simple graded Lie algebras found by \textit{E. I. Zel'manov} [Mat. Sb., Nov. Ser. 124(166), No. 3(7), 353-392 (1984; Zbl 0546.17005)]\ to show that the Lie algebras treated there have 3-grading or a 5-grading.
scientific article; zbMATH DE number 1089119
scientific article