Abstract
Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
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Acknowledgements
This work was done while the first author visited Linköping’s University. He would like to thank the Mathematical Institution of Linköping’s University for hospitality. The second author was partially supported by G.S. Magnusons Foundation, grant MG 2017-0101. We also would like to thank the anonymous referee for his/her detailed comments and suggestions.
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This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.
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Krasnov, Y., Tkachev, V.G. Idempotent Geometry in Generic Algebras. Adv. Appl. Clifford Algebras 28, 84 (2018). https://doi.org/10.1007/s00006-018-0902-7
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DOI: https://doi.org/10.1007/s00006-018-0902-7