Year: 2026
Type: preprint
Abstract: This paper introduces and studies $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{2})^3 - (x^{3})^2 = 0$. This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree evanescent identities in degree $6$, and encompasses several well-studied structures, including Jordan and medial algebras. The primary motivation for investigating these algebras lies in their trivial Peirce polynomials, which removes a priori restrictions on the spectrum of the multiplication operator associated with an idempotent. This paper reviews and further develops the theory of Peirce operators, Peirce polynomials, and second-order linearizations. It shows that despite the triviality of the Peirce polynomial, any idempotent $c$ admits well-behaved, explicit fusion rules for multiplication between its $λ$-Peirce spaces for $λ\neq \tfrac{1}{2}$. Furthermore, multiplication by a nonsingular idempotent is an algebra homomorphism. Finally, there are given concrete examples of $(2,3)$-palintropic algebras, together with applications of these algebraic structures to commutative polynomial maps.
Source: arXiv (Cornell University)
Authors: Fox, Daniel J. F., Tkachev, Vladimir G.
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166506958