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Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras

  • Alberto Elduque,
    Departamento de Matemáticas, Universidad de Zaragoza , 50009 Zaragoza, Spain
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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 symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
MSC Classifications: 17A75, 14L15, 17B25, 20G15 show english descriptions Composition algebras
Group schemes
Exceptional (super)algebras
Linear algebraic groups over arbitrary fields
17A75 - Composition algebras
14L15 - Group schemes
17B25 - Exceptional (super)algebras
20G15 - Linear algebraic groups over arbitrary fields
 

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