Coordinatization Theorems For Graded Algebras

Abstract

AbstractIn this paper we study simple associative algebras with finite-gradings. This is done using a simple algebraFgthat has been constructed in Morita theory from a bilinear formg:U×V→Aover a simple algebraA. We show that finite-gradings onFgare in one to one correspondence with certain decompositions of the pair (U, V). We also show that any simple algebraRwith finite-grading is graded isomorphic toFgfor some bilinear fromg:U×V→A, where the grading onFgis determined by a decomposition of (U, V) and the coordinate algebraAis chosen as a simple ideal of the zero componentR0ofR. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.