Adjoint Representation [The Physics Travel Guide]

--- advanced_tools:group_theory:representation_theory:adjoint_representation [2018/04/15 16:34]
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 <tabbox Concrete>
+  * 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}}]
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 </note>
 <tabbox Why is it interesting?>
+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?
+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!
+The representation of each group where it acts on its own Lie algebra is called the adjoint representation.