Is there a standard notation for the unit quaternions?

Question

Of course, I mean other than $SU(2)$. I think the standard notation for the entire algebra of quaternions is $\mathbb{H}$, so I would imagine something like $\mathbb{H}^*$ or $\mathbb{H}_1$, etc. Is there any at all in the literature?

Answer 1

Yes, there is a standard notation for the unit quaternions. In this notation, a unit quaternion is typically represented as $q = a + bi + cj + dk$, where $a, b, c$, and d are real numbers and $i, j,$ and $k$ are the quaternion units.

The set of quaternions is denoted as $\mathbb H = \{a + bi + cj + dk : a, b, c, d ∈ ℝ\} $

In notation, $\mathbb H₁ = \{q ∈ H : |q| = 1\}$, where $|q|$ denotes the norm (magnitude) of the quaternion $q$.

Answer 2

I don't think there's actually an enforced/standardized/normed notation for this.

Depending on where you look you will find $S^{3}$ (unit $3$-Sphere), $\mathbb{H}^{*}$, $\operatorname{SU}\left( 2 \right)$ (Special Unitary Group), ...

Unit $3$-Sphere around coordinate origin

A $3$-sphere with center at the coordinate origin and radius $1$ is $S^{3}$. The equation that results is (or for a $n$-Sphere: $S^{n}$): $$ \begin{align*} S^{n} &= \left\{ \left( x_{0},\, x_{1},\, \dots,\, x_{n} \right) \in \mathbb{R}^{n + 1}{:}\, \sum\limits_{k = 0}^{n}\left[ x_{k}^{2} \right] = 1 \right\}\\ S^{3} &= \left\{ \left( x_{0},\, x_{1},\, x_{2},\, x_{3} \right) \in \mathbb{R}^{4}{:}\, \sum\limits_{k = 0}^{4}\left[ x_{k}^{2} \right] = 1 \right\}\\ \end{align*} $$

We can do this using the rule $\left| z \right| = \sum\limits_{k = 0}^{n}\left[ x_{k}^{2} \right]$ (with $z$ as a hypercomplex number of dimension $n$ and the parts $x_{0 }$, $x_{1}$, ..., $x_{n}$) rewrite to: $$ \begin{align*} S^{n} &= \left\{ \left( x_{0},\, x_{1},\, \dots,\, x_{n} \right) \in \mathbb{R}^{n + 1}{:}\, \sum\limits_{k = 0}^{n}\left[ x_{k}^{2} \right] = 1 \right\}\\ S^{n} &= \left\{ z \in \mathbb{R}^{n + 1}{:}\, \left| z \right| = 1 \right\}\\ \\ S^{3} &= \left\{ \left( x_{0},\, x_{1},\, x_{2},\, x_{3} \right) \in \mathbb{R}^{4}{:}\, \sum\limits_{k = 0}^{4}\left[ x_{k}^{2} \right] = 1 \right\}\\ S^{3} &= \left\{ \left( z_{1},\, z_{2} \right) \in \mathbb{C}^{2}{:}\, \sum\limits_{k = 1}^{2}\left[ \left| z_{k} \right|^{2} \right] = 1 \right\}\\ S^{3} &= \left\{ q \in \mathbb{H}{:}\, \left| q \right|^{2} = 1 \right\}\\ \end{align*} $$

So $$\fbox{$ \begin{align*} S^{3} = \left\{ q \in \mathbb{H}{:}\, \left| q \right| = 1 \right\}\\ \end{align*} $}$$ is the set of all Unit Quaternions.

You often find this notation in topology or geometry. $S^{3}$ is also isomorphic to the groups $\operatorname{Spin}\left( 3 \right)$ and $\operatorname{SU}\left( 2 \right)$,

Quaternion-star ($\mathbb{H}^{*}$)

So far I have only seen $\mathbb{H}^{*}$ for non-zero quaternions ($\mathbb{H}^{*} = \left\{ q \in \mathbb{H} \mid q \ne 0 \right\}$) and so far only encountered it in algebra and proofs.

Special Unitary Group ($\operatorname{SU}\left( 2 \right)$) and Compact Classical Group ($\operatorname{U}\left( 2 \right)$)

By definition $\operatorname{SU}\left( 2 \right) \subseteq \operatorname{U}\left( 2 \right)$.

Using matrix multiplication for the binary operation, $\operatorname{SU}\left( 2 \right)$ forms a group: $$ \begin{align*} \operatorname{SU}\left( 2 \right) &= \left\{ \begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}{:}\, \left\{ \alpha,\, \beta \right\} \in \mathbb{C},\, \left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1 \right\}\\ \end{align*} $$

We know that $\begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}$ is a possible matrix representation of quaternions, which directly follows from the Cayley-Dickson Construction and $\left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1 = \left| q \right|$ makes it a unit quaternion: $$ \begin{align*} \operatorname{SU}\left( 2 \right)~ &\widehat{=} \left\{ \alpha + \beta \cdot j{:}\, \left\{ \alpha,\, \beta \right\} \in \mathbb{C},\, \left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1 \right\}\\ \operatorname{SU}\left( 2 \right)~ &\widehat{=} \left\{ q{:}\, q \in \mathbb{H},\, \left| q \right|^{2} = 1 \right\}\\ \operatorname{SU}\left( 2 \right)~ &\widehat{=} \left\{ q \in \mathbb{H}{:}\, \left| q \right| = 1 \right\}\\ \end{align*} $$