advanced_tools:group_theory:representation_theory:adjoint_representation

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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 [2020/12/26 22:49] edi [Concrete] |
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* 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 | ||

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- | [{{ :advanced_tools:group_theory:representation_theory:adjointaction.png?nolink |Diagram by Eduard Sackinger}}] | + | **Example** |

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+ | 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 act like $L'=[J,L]$. It is possible to rewrite the adjoint representation such that it acts on 3-dimensional vectors (as opposed to 2x2 matrices) by regular matrix-vector multiplication. | ||

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+ | [{{ :advanced_tools:group_theory:representation_theory:su2_adjoint.jpg?nolink }}] | ||

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+ | For a more detailed explanation of this diagram as well as adjoint representations of other groups, see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||

<tabbox Abstract> | <tabbox Abstract> | ||

advanced_tools/group_theory/representation_theory/adjoint_representation.txt · Last modified: 2020/12/26 22:49 by edi

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