advanced_tools:group_theory:sp_1

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+ | ====== Sp(1) ====== | ||

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+ | <tabbox Intuitive> | ||

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+ | $Sp(1)$ 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.) | ||

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

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+ | <tabbox Concrete> | ||

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+ | **Representations** | ||

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+ | 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]]. | ||

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+ | [{{ :advanced_tools:group_theory:representation_theory:sp1_quaternions.jpg?nolink }}] | ||

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+ | <tabbox Abstract> | ||

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+ | <note tip> | ||

+ | The motto in this section is: //the higher the level of abstraction, the better//. | ||

+ | </note> | ||

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+ | <tabbox Why is it interesting?> | ||

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+ | /*<tabbox FAQ>*/ | ||

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+ | /*<tabbox History>*/ | ||

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+ | </tabbox> | ||

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advanced_tools/group_theory/sp_1.txt ยท Last modified: 2020/12/12 23:37 by edi

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