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Order 3 Elements in G2 and Idempotents in Symmetric Composition Algebras

Published online by Cambridge University Press:  20 November 2018

Alberto Elduque
Alberto Elduque*
Affiliation:
Departamento de Matemácas, Universidad de Zaragoza, 50009 Zaragoza, Spain, e-mail: elduque@unizar.es
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Abstract

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Order three elements in the exceptional groups of type ${{G}_{2}}$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.

Over an algebraically closed field, there are two conjugacy classes of order three elements in ${{G}_{2}}$ in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Keywords

17A7514L1517B2520G15symmetric composition algebraOkubo algebraautomorphism groupcentralizeridempotent
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 1038 - 1075
Copyright
Copyright © Canadian Mathematical Society 2018

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