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 advanced_tools:group_theory:representation_theory:adjoint_representation [2020/11/29 16:24]edi [Intuitive] advanced_tools:group_theory:representation_theory:adjoint_representation [2020/11/29 16:59]edi [Concrete] Both sides previous revision Previous revision 2020/12/26 22:49 edi [Concrete] 2020/11/29 16:59 edi [Concrete] 2020/11/29 16:24 edi [Intuitive] 2018/04/15 16:38 aresmarrero [Concrete] 2018/04/15 16:38 aresmarrero [Why is it interesting?] 2018/04/15 16:35 aresmarrero [Concrete] 2018/04/15 16:34 aresmarrero created 2020/12/26 22:49 edi [Concrete] 2020/11/29 16:59 edi [Concrete] 2020/11/29 16:24 edi [Intuitive] 2018/04/15 16:38 aresmarrero [Concrete] 2018/04/15 16:38 aresmarrero [Why is it interesting?] 2018/04/15 16:35 aresmarrero [Concrete] 2018/04/15 16:34 aresmarrero created Last revision Both sides next revision Line 9: Line 9:  ​  ​ * For a detailed discussion, see [[http://​jakobschwichtenberg.com/​adjoint-representation/​|What’s so special about the adjoint representation of a Lie group?]] by J. Schwichtenberg * For a detailed discussion, see [[http://​jakobschwichtenberg.com/​adjoint-representation/​|What’s so special about the adjoint representation of a Lie group?]] by J. Schwichtenberg - ---- ---- - [{{ :​advanced_tools:​group_theory:​representation_theory:​adjointaction.png?​nolink ​|Diagram by Eduard Sackinger}}] + **Example** - + + The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the adjoint representation,​ we use the Lie algebra as the representation space, as shown in the lower branch (red arrows). The group elements act on this space like $L'​=ULU^{-1}$ and the Lie-algebra elements like $L'​=[J,​L]$. It is possible to rewrite this representation such that it acts on 3-dimensional vectors (as opposed to 3x3 matrices) by regular matrix-vector multiplication. + + [{{ :​advanced_tools:​group_theory:​representation_theory:​su2_adjoint.jpg?nolink }}] + + For a more detailed explanation of this diagram as well as adjoint representations of other groups, see [[https://​esackinger.wordpress.com/​|Fun with Symmetry]]. ​  ​  ​