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advanced_tools:group_theory:representation_theory:adjoint_representation [2018/04/15 16:35] aresmarrero [Concrete] |
advanced_tools:group_theory:representation_theory:adjoint_representation [2018/04/15 16:38] aresmarrero [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}}] | [{{ :advanced_tools:group_theory:representation_theory:adjointaction.png?nolink |Diagram by Eduard Sackinger}}] | ||
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</note> | </note> | ||
- | <tabbox Why is it interesting?> | + | <tabbox Why is it interesting?> |
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+ | A [[advanced_tools:group_theory:representation_theory|representation]] is a map that maps each element of the set of abstract groups element to a matrix that acts on a vector space. A confusing point here is: If we can study the representation of any group on any vector space, where should we start? | ||
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+ | Luckily there comes exactly only distinguished vector space automatically with each Lie group: the [[advanced_tools:group_theory:lie_algebras|Lie algebra]] of the group! | ||
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+ | The representation of each group where it acts on its own Lie algebra is called the adjoint representation. | ||
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