Evaluations of multilinear polynomials on low rank Jordan algebras

TL;DR

This paper proves the generalized Kaplansky conjecture for specific Jordan algebras, showing that the image of multilinear polynomials is limited to certain subspaces, thus advancing understanding of polynomial behavior in algebraic structures.

Contribution

It establishes the generalized Kaplansky conjecture for Jordan algebras of type J_n, including self-adjoint matrices over various fields, identifying possible polynomial images.

Findings

The image of multilinear polynomials is either {0}, R, V, or J_n.

The result applies to self-adjoint 2x2 matrices over R, C, H, and Oct.

Provides a classification of polynomial images in these Jordan algebras.

Abstract

In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of multilinear polynomial must be either $\{0\}$, $\mathbb{R}$, the space of pure elements $V$, or $J_n$.