In Brian Hall's Book "Lie Groups, Lie Algebras and Representations" I find the following construction.
Let $J \colon \mathbb{C}^{2n} \longrightarrow \mathbb{C}^{2n}$ be a map defined as
$$J(\alpha,\beta) = (-\beta^*, \alpha^*)$$
where $(\alpha,\beta) \in \mathbb{C}^{2n}$ and $^*$ stands for complex conjugation. Observe that $J$ is not linear but is conjugate-linear.
Define moreover the operators $i \colon \mathbb{C}^{2n} \longrightarrow \mathbb{C}^{2n}$ (scalar multiplication by the complex number i) and $K = iJ$.
By putting
$$\mathbf{i} = i \quad \mathbf{j} = J \quad \mathbf{k} = K$$
and noting that these operators, together with the identity $I_{\mathbb{C}^{2n}}$, obey the quaternion multiplication rules, we have realized the quaternion algebra. In the book is stated that
$$\mbox{"We can therefore make $\mathbb{C}^{2n}$ into a vector space over the noncommutative algebra $\mathbb{H}$"}$$
- What does this mean exactly? I understand the construction, but would
rather say that the quaternions have been represented over a certain
set of operators acting on $\mathbb{C}^{2n}$.
Best Answer
As you say, the definition of $I$, $J$ and $K$ give a representation of $\mathbb{H}$ on $\mathbb{C}^{2n}$. This makes $\mathbb{C}^{2n}$ into an $\mathbb{H}$-module. Thus, $\mathbb{C}^{2n}$ can be considered as a quaternionic vector space, the word vector space is used because $\mathbb{H}$ is a division ring.