User Tools

Site Tools


advanced_tools:group_theory:representation_theory:adjoint_representation

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
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]
Line 9: Line 9:
 <tabbox Concrete> ​ <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   * 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 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. 
 + 
 +[{{ :​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]]. ​
 <tabbox Abstract> ​ <tabbox Abstract> ​
  
advanced_tools/group_theory/representation_theory/adjoint_representation.txt · Last modified: 2020/12/26 22:49 by edi