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Coordinatization Theorems For Graded Algebras

Published online by Cambridge University Press:  20 November 2018

Bruce Allison  and
Oleg Smirnov
Bruce Allison
Affiliation:
Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta T6G 2G1, e-mail: ballison@math.ualberta.ca
Oleg Smirnov
Affiliation:
Department of Mathematics College of Charleston Charleston, South Carolina 29424-0001 USA, e-mail: smirnov@cofc.edu
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Abstract

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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.

Keywords

16W50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 4 , 01 December 2002 , pp. 451 - 465
Copyright
Copyright © Canadian Mathematical Society 2002

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