The Free Hamilton Algebra

TL;DR

This work analyzes the free Hamilton algebra $\mathcal{W}_{p,q}$, the free product of two $2$-dimensional algebras over a field $\mathbb{F}$, by exploiting connections to quaternion algebras. It develops a Clifford-like framework centered on the adjunction, the central element $\omega$, and a trace/inner-product structure valued in $\mathbb{F}[\omega]$, enabling concrete criteria for zero divisors, units, and finite-dimensional subalgebras. The Zero Divisors Theorem characterizes zero divisors via the splitting of $p$ or $q$ and is complemented by new embedding results into matrix algebras; the Weak Units Theorem and a detailed analysis of the unit group illuminate the structure of monomial vs non-monomial units. The paper then provides a thorough classification of maximal ideals, constructs and classifies finite-dimensional subalgebras, and finally proves an Automorphisms Theorem: every automorphism of $\mathcal{W}_{p,q}$ splits uniquely as a basic automorphism followed by an inner automorphism, with a comprehensive analysis of the $C$-automorphism subgroup, signatures, and case-by-case behavior depending on the splitting structure of $p$ and $q$. The results reveal a close link to quaternion algebras (global and local), and give a complete structural picture of $\mathcal{W}_{p,q}$ including its center, unit group, and automorphism group, thereby advancing understanding of free products of $2$-dimensional algebras and their representations.

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.

Theorems & Definitions (236)

• Theorem 1.1: Zero Divisors Theorem
• Theorem 1.2: Weak Units Theorem
• Theorem 1.3: Strong Units Theorem
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10: Automorphisms Theorem