Commutative algebras satisfying univariate identities with vanishing Peirce polynomial
Abstract
This paper introduces and studies (2,3)-palintropic algebras, a class of commutative algebras defined by the identity (x2)3 − (x3)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≠ 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.
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BibTeX
@misc{fox2026commutative,
title = {{Commutative algebras satisfying univariate identities with vanishing Peirce polynomial}},
author = {Daniel J. F. Fox and Vladimir G. Tkachev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.27496},
eprint = {2606.27496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27496},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Commutative algebras satisfying univariate identities with vanishing Peirce polynomial
AU - Daniel J. F. Fox
AU - Vladimir G. Tkachev
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2606.27496
UR - https://arxiv.org/abs/2606.27496
AB - 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≠ 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.
ER -