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advanced_tools:group_theory:central_extension
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Central Extension
Why is it interesting?
Central extensions are a standard trick to convert projective representations of some group into true representations of another group.
This is necessary, because when we only consider the "naive" normal representations of a group like the Lorentz group, we miss an important representation (the spin $\frac{1}{2}$) representation). Thus, we can either use a less restrictive definition of a representation, i.e. use projective representations instead of true representations, or we could simply work with true representations of the central extension of the given group.
For example, the projective representations of $SO(3,1)$ correspond to regular representations of $SL(2,\mathbb{C})$.
Layman
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.
Student
The central extension $\hat G$ of a given group $G$ by an abelian group $A$ is defined as a group such that $A$ is a subgroup of the center of $\hat G$ and that the quotient $\hat G/A = G$.
- See page 178 in Moonshine beyond the Monster by Terry Gannon
Researcher
The motto in this section is: the higher the level of abstraction, the better.
Examples
- Example1
- Example2:
FAQ
History
advanced_tools/group_theory/central_extension.1513509098.txt.gz · Last modified: 2017/12/17 11:11 (external edit)
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