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 — advanced_tools:group_theory:sp_1 [2020/12/12 23:37] (current)edi created 2020/12/12 23:37 edi created 2020/12/12 23:37 edi created Line 1: Line 1: + ====== Sp(1) ====== + +  ​ + + $Sp(1)$ 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 $SU(2)$. ​ + ​ +  ​ + + **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 }}] + +  ​ + + + The motto in this section is: //the higher the level of abstraction,​ the better//. + ​ + +  ​ + + /​*<​tabbox FAQ>​*/ ​ + + /​*<​tabbox History>​*/ ​ + + ​ +