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A correction of the decomposability result in a paper by Meyer-Neutsch
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-8422-6140
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 504, p. 432-439Article in journal (Refereed) Published
Abstract [en]

In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra G. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch’s paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation L : x → ax) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product.

Place, publisher, year, edition, pages
2018. Vol. 504, p. 432-439
Keywords [en]
Commutative nonassociative algebras, Griess algebra, Idempotents, Associative bilinear form, Metrised algebras
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:liu:diva-145611DOI: 10.1016/j.jalgebra.2018.02.031ISI: 000431284200013OAI: oai:DiVA.org:liu-145611DiVA, id: diva2:1188675
Available from: 2018-03-08 Created: 2018-03-08 Last updated: 2018-05-31

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The full text will be freely available from 2020-03-07 16:10
Available from 2020-03-07 16:10

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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf