User Tools

Site Tools


advanced_tools:group_theory:sp_1

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

advanced_tools:group_theory:sp_1 [2020/12/12 23:37] (current)
edi created
Line 1: Line 1:
 +====== 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/​|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>​
 +
  
advanced_tools/group_theory/sp_1.txt ยท Last modified: 2020/12/12 23:37 by edi