The toral Tits process of Jordan algebras
type: Item
instance of: scholarly article
title: The toral Tits process of Jordan algebras
zbMATH Open document ID: 0552.17013
DOI: 10.1007/BF02941454
author: Holger P. Petersson
author: Michel L. Racine
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1984-01-01T00:00:00Z
review text: The paper is a continuation of the authors' work on exceptional simple Jordan algebras. The results obtained in the authors' papers ''Jordan algebras of degree 3 and the Tits process'' and ''Classification of algebras arising from the Tits process'', J. Algebra (to appear) show that 9-dimensional subalgebras of exceptional simple Jordan algebras are of special interest. In the authors' work such algebras have the form \(E\oplus (E\otimes C)\) for two separable commutative and associative algebras E,C of dimension 2,3 respectively, and the multiplication is described using the general Tits process. On the other hand, by structure theory such algebras are either of the form \(A^+\) or H(B,*) where A,B are simple associative algebras of degree 3 and * is an involution of the second kind. It is therefore a natural objective to derive a criterion for deciding when an algebra \(E\oplus (E\otimes C)\) is isomorphic to an \(A^+\) or H(B,*) respectively. The authors show: \(E\oplus (E\otimes C)\cong A^+\) if and only if \(\Delta (E)=\delta (C)\) where \(\delta\) (E) resp. \(\delta\) (C) are the discriminants which the authors associated to E and C in [Manuscr. Math. 45, 249-272 (1984; Zbl 0536.17009)]. The authors' criterion follows from a description of the center of the special universal envelope of \(E\oplus (E\otimes C)\).
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 17C40
zbMATH DE Number: 3879115
zbMATH Keywords: toral Tits process
zbMATH Keywords: exceptional simple Jordan algebras
zbMATH Keywords: 9-dimensional subalgebras
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02941454
OpenAlex ID: W2076484552
description: scientific article; zbMATH DE number 3879115