Variety of Idempotents in Nonassociative Algebras

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 − 1 nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension n. We also discuss the exceptionality of the eigenvalue λ = 1 2 which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.

V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras

“…The ultimate classification of nonassocaitive algebras of cubic minimal cones is still an incomplete project but a complete picture seems clear. A interesting and deep direction here is a better understanding of general metrisable algebras and their Peirce structure in different differential geometric and group theoretic contexts [18], [8], [10], [16].…”

“…Note also that the Peirce value 1 2 appears and plays a crucial role in the classification of nonassociative algebras associated with Hsiang exceptional eigencubics. In fact, this Peirce number is remarkable in many aspects and indicates that an algebra must satisfy to a specific algebra identity, see [56], [18], [56].…”

“…By using the tetrad decomposition, one can show that some Peirce dimensions in the above table are not realizable. This eliminates the Perice triples (n 1 , n 2 , n 3 ) with values (2, 5, 7), (2,8,10), (2,14,16), (3,14,18), (0, 26, 24), (2,26,28), (3,26,30) and (7,26,38). All triples with n 2 = 0, n 1 = 0 and n 2 = 26, n 1 = 1 and (n 1 , n 2 , n 3 ) = (4, 5, 11) are realizable, see the description below:…”

“…The interested reader is referred to the recent monograph [31] for unexpected connections of nonassociative algebras to regularity theory of fully nonlinear PDEs. See also the recent papers [52], [55], [56], [18], [58], [57] for further results and basic concepts considered below.…”

Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs

“…To give the latter observation a rigorous meaning we need to formalize what we mean by a generic cubic form. One natural way to do this is to combine the correspondence given in the proceeding section with the concept of a generic algebra introduced recently in [25]. Recall that an algebra over C is called generic if it has exactly 2 n idempotents (counting x = 0).…”