====== Sp(1) ======

<tabbox 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 $SU(2)$.   
  
<tabbox 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 [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#sp1_quaternion|Fun with Symmetry]].

[{{ :advanced_tools:group_theory:representation_theory:sp1_quaternions.jpg?nolink }}]
 
<tabbox Abstract> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox Why is it interesting?>   

/*<tabbox FAQ>*/ 

/*<tabbox History>*/ 

</tabbox>