Coordinatization Theorems For Graded Algebras

Allison, Bruce ; Smirnov, Oleg

Canadian mathematical bulletin, Tome 45 (2002) no. 4, pp. 451-465

Résumé

In this paper we study simple associative algebras with finite $\mathbb{Z}$ -gradings. This is done using a simple algebra ${{F}_{g}}$ that has been constructed in Morita theory from a bilinear form $g:\,U\,\times \,V\,\to \,A$ over a simple algebra $A$ . We show that finite $\mathbb{Z}$ -gradings on ${{F}_{g}}$ are in one to one correspondence with certain decompositions of the pair $\left( U,\,V \right)$ . We also show that any simple algebra $R$ with finite $\mathbb{Z}$ -grading is graded isomorphic to ${{F}_{g}}$ for some bilinear from $g:\,U\,\times \,V\,\to \,A$ , where the grading on ${{F}_{g}}$ is determined by a decomposition of $\left( U,\,V \right)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component ${{R}_{0}}$ of $R$ . In order to prove these results we first prove similar results for simple algebras with Peirce gradings.

Détail

DOI : 10.4153/CMB-2002-048-4

Mots-clés : 16W50