The Free Hamilton Algebra

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.