====== Lie Groups ======

<tabbox Why is it interesting?> 

Lie groups describe continuous symmetry and lie at the heart of modern physics. For example, the symmetry group of the [[models:standard_model|standard model of particle physics]] and the best spacetime symmetry group that we know (the Poincare group) are Lie groups. 

<tabbox Layman> 

A Lie group is a continuous set of transformations that satisfy the [[advanced_tools:group_theory:|group axioms]]. A good example for a Lie group is the symmetry group of the circle. A rotation by $5^\circ$ is a symmetry of the circle and a rotation by $0.00001^\circ$ is a symmetry, too. In contrast, the symmetry group of a square is not continuous. A rotation by $90^\circ$ is a symmetry, whereas a rotation by $5^\circ$ is not a symmetry.
  
<tabbox Student> 
The diagram below shows some low-dimensional, but important, Lie groups and their relationships.

[{{ :advanced_tools:group_theory:group_relations.jpg?nolink }}]

For a more detailed explanation of this diagram and much more, see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#group_relations|Fun with Symmetry]].

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For book recommendations, see [[advanced_tools:group_theory|here]]. 
 
<tabbox Researcher> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
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<tabbox Examples> 

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<tabbox FAQ> 
  
<tabbox History> 

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