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# Order $3$ Elements in $G_2$ and Idempotents in Symmetric Composition Algebras

Published:2017-12-19
Printed: Oct 2018
• Alberto Elduque,
Departamento de Matemáticas, Universidad de Zaragoza , 50009 Zaragoza, Spain
 Format: LaTeX MathJax PDF

## Abstract

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: symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
 MSC Classifications: 17A75 - Composition algebras 14L15 - Group schemes 17B25 - Exceptional (super)algebras 20G15 - Linear algebraic groups over arbitrary fields

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