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.
Highlights
The aim of this paper is to further develop the geometrical approach to generic commutative algebras initiated in [8] and based on the topological index theory and singularity theory
We are interested in the following question: How the geometry of idempotents in a generic algebra is determined by its algebraic structure, and vice versa
It is classically known that idempotents play a distinguished role in associative and nonassociative (Jordan, octonions) algebra structures [12,16]; see very recent results for the so-called axial algebras [6] and nonassociative algebras of cubic minimal cones [14,17,18]
Summary
The aim of this paper is to further develop the geometrical approach to generic commutative (maybe nonassociative) algebras initiated in [8] and based on the topological index theory and singularity theory. We are interested in the following question: How the geometry of idempotents in a generic algebra is determined by its algebraic structure, and vice versa. A key ingredient of our approach in [8] is the Euler-Jacobi formula which gives an algebraic relation between the critical points of a polynomial map and their indices. There is a natural bijection between fixed points of the squaring map ψ : x → x2 in a nonassociative algebra A and its idempotents. 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. ∗Corresponding author
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