Sp(1)

Prerequisites: Quaternions, Lie Groups. Full path (5 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

$Sp(1)$ is the group of quaternions of unit length, also known as the compact symplectic group of order one, hence the name $Sp(1)$. (See https://math.ucr.edu/home/baez/symplectic.html for an explanation of the name.)

The group $Sp(1)$ is isomorphic to $SU(2)$.

Concrete

Representations

The diagram below shows the defining representation of $Sp(1)$ in its upper branch and the (3-dimensional) adjoint representation of the same group in the lower branch. The latter representation can be used to rotate objects in 3D space and has applications in computer graphics. For a more detailed explanation of this diagram and representations of other Lie groups see Fun with Symmetry.

Contributing authors:

Abstract

Write a quaternion as $q = a + b\,i + c\,j + d\,k$ with $i^2=j^2=k^2=ijk=-1$ and conjugate $\bar q = a - b\,i - c\,j - d\,k$. The norm $|q|^2 = q\bar q = a^2+b^2+c^2+d^2$ is multiplicative, $|pq| = |p||q|$, so the unit quaternions form a group $$Sp(1) = \{q\in\mathbb{H} : |q| = 1\}\cong S^3 .$$ It is compact, connected and simply connected.

Isomorphism with $SU(2)$. The map $$a + b\,i + c\,j + d\,k\ \longmapsto\ \begin{pmatrix} a + ib & c + id\\ -c + id & a - ib\end{pmatrix}$$ sends $i\mapsto i\sigma_3$, $j\mapsto i\sigma_2$, $k\mapsto i\sigma_1$, respects products, and maps unit quaternions onto matrices of the form $\begin{pmatrix}\alpha & \beta\\ -\bar\beta & \bar\alpha\end{pmatrix}$ with $|\alpha|^2+|\beta|^2=1$. These are exactly the elements of $SU(2)$.

Lie algebra and rotations. The tangent space at $1$ is the space of imaginary quaternions $\mathrm{Im}\,\mathbb{H}\cong\mathbb{R}^3$. For imaginary $u,v$ we have $uv = -\vec u\cdot\vec v + \vec u\times\vec v$, so the bracket is $$[u,v] = uv - vu = 2\,\vec u\times\vec v ,$$ and $\mathfrak{sp}(1)\cong\mathfrak{so}(3)$. The adjoint action $v\mapsto qv\bar q$ on imaginary quaternions preserves the norm and orientation, which gives the double cover $Sp(1)\to SO(3)$ with kernel $\{\pm 1\}$. The unit quaternion $q = \cos\frac{\theta}{2} + \sin\frac{\theta}{2}\,\hat n$ (with $\hat n$ an imaginary unit quaternion) produces the rotation by $\theta$ about $\hat n$. Similarly $(p,q)\mapsto(x\mapsto px\bar q)$ on $\mathbb{H}\cong\mathbb{R}^4$ gives $Sp(1)\times Sp(1)\cong\mathrm{Spin}(4)$, a double cover of $SO(4)$.

General $n$. $Sp(n)$ is the group of $n\times n$ quaternionic matrices that preserve the standard quaternionic Hermitian form, equivalently $Sp(n) = Sp(2n,\mathbb{C})\cap U(2n)$. It is the compact real form of the Lie algebra of type $C_n$. Besides $Sp(1)\cong SU(2)$, there is the low-dimensional isomorphism $Sp(2)\cong\mathrm{Spin}(5)$. The compact group $Sp(1)$ should not be confused with the non-compact real symplectic group $Sp(2,\mathbb{R})\cong SL(2,\mathbb{R})$; the two are different real forms of the same complex group $Sp(2,\mathbb{C})\cong SL(2,\mathbb{C})$.

Path to this page

1. Imaginary Numbers
2. Linear Algebra
3. Group Theory
4. Quaternions
5. Lie Groups

Authors: Eduard Sackinger