The Free Hamilton Algebra

TL;DR

This paper studies the structure of free Hamilton algebras over arbitrary fields, revealing their connections to quaternion algebras and analyzing key algebraic properties such as zero divisors, units, ideals, subalgebras, and automorphisms.

Contribution

It explores the algebraic structure of free Hamilton algebras, leveraging their connection to quaternion algebras to analyze fundamental properties and symmetries.

Findings

Identified conditions for zero divisors in free Hamilton algebras

Characterized the automorphism group of these algebras

Described the maximal ideals and finite-dimensional subalgebras

Abstract

Over an arbitrary field $\mathbb{F}$, let $p$ and $q$ be monic polynomials with degree $2$ in $\mathbb{F}[t]$. The free Hamilton algebra of the pair $(p,q)$ is the free noncommutative algebra in two generators $a$ and $b$ subject only to the relations $p(a)=0=q(b)$. Free Hamilton algebras are models of free products of two $2$-dimensional algebras over $\mathbb{F}$. They can be viewed as the most elementary nontrivial noncommutative algebras over fields. It has been recently observed that the free Hamilton algebra has surprising connections with quaternion algebras. Here, we exploit these connections to investigate its zero divisors, group of units, maximal ideals, finite-dimensional subalgebras, and its automorphism group.