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Variety of Idempotents in Nonassociative Algebras

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Topics in Clifford Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least n 2 − 1 nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension n. We also discuss the exceptionality of the eigenvalue \(\lambda =\frac 12\) which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrized algebras.

Dedicated to Professor Wolfgang Sprößig on the occasion of his 70th birthday

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Notes

  1. 1.

    A three dimensional generic algebra may a priori have 14 = (23 − 1) × (3 − 1) distinct eigenvalues except 0 and 1.

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Acknowledgements

V. Tkachev has been partially supported by Stiftelsen GS Magnusons fond, grant MG2018-0042. The authors are very grateful to the reviewer for his/her careful and reading of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Yakov Krasnov

  2. Department of Mathematics, Linköping University, Linköping, Sweden

    Vladimir G. Tkachev

Authors
  1. Yakov Krasnov
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  2. Vladimir G. Tkachev
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Correspondence to Yakov Krasnov .

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Editors and Affiliations

  1. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Swanhild Bernstein

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Krasnov, Y., Tkachev, V.G. (2019). Variety of Idempotents in Nonassociative Algebras. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_20

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