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advanced_tools:group_theory:representation_theory [2018/04/09 15:57] tesmitekle |
advanced_tools:group_theory:representation_theory [2018/04/09 15:58] tesmitekle [Intuitive] |
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<tabbox Intuitive> | <tabbox Intuitive> | ||
- | + | <blockquote>geometry asks, “Given a geometric object X, what is its group of | |
- | <note tip> | + | symmetries?” Representation theory reverses the question to “Given a group G, what objects X |
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | does it act on?” and attempts to answer this question by classifying such X up to isomorphism." [[https://math.berkeley.edu/~teleman/math/RepThry.pdf|Source]]</blockquote> |
- | </note> | + | |
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<tabbox Concrete> | <tabbox Concrete> | ||
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- | <WRAP tip>**Basic idea:** | ||
- | "//geometry asks, “Given a geometric object X, what is its group of | ||
- | symmetries?” Representation theory reverses the question to “Given a group G, what objects X | ||
- | does it act on?” and attempts to answer this question by classifying such X up to isomorphism.//" [[https://math.berkeley.edu/~teleman/math/RepThry.pdf|Source]] </WRAP> | ||
A Lie group is in abstract terms a manifold, which obeys the group axioms. A **representation** is a special type of map $R$ from this manifold to the linear operators of some vector space. The map must obey the condition | A Lie group is in abstract terms a manifold, which obeys the group axioms. A **representation** is a special type of map $R$ from this manifold to the linear operators of some vector space. The map must obey the condition |