Rota-Baxter operators of zero weight on simple Jordan algebra of Clifford type

TL;DR

This paper classifies Rota-Baxter operators of zero weight on simple Jordan algebras of Clifford type, establishing their nilpotency properties and exact indices over various fields using algebraic and number-theoretic methods.

Contribution

It provides a complete characterization of nilpotent Rota-Baxter operators of zero weight on these algebras, including explicit nilpotency indices over different fields.

Findings

Rota-Baxter operators are nilpotent of index ≤ 3 on these algebras.

Exact nilpotency indices are determined over , , and  fields.

Number theory techniques are used to analyze operators over finite fields.

Abstract

It is proved that any Rota---Baxter operator of zero weight on Jordan algebra of a nondegenerate bilinear symmetric form is nilpotent of index less or equal three. We state exact value of nilpotency index on simple Jordan algebra of Clifford type over fields $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Z}_p$. For $\mathbb{Z}_p$, we essentially use the results from number theory concerned quadratic residues and Chevalley---Warning theorem.