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--- advanced_tools:group_theory:central_extension [2017/12/17 11:26]
+++ advanced_tools:group_theory:central_extension [2017/12/17 12:12]
jakobadmin
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+====== Central Extension ======
+<tabbox 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 $SL(2,\mathbb{C})$.
+<tabbox Layman>
+<note tip>
+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.
+</note>
+<tabbox Student>
+<WRAP tip> 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 $\hat G/A = G$. </WRAP>
+  * See page 178 in Moonshine beyond the Monster by Terry Gannon
+<tabbox Researcher>
+<note tip>
+The motto in this section is: //the higher the level of abstraction, the better//.
+</note>
+<tabbox Examples>
+--> Galilean group -> Bargmann group#
+The classical Galilean group needs to be extended by the introduction of a central charge, called //mass//, and this yields the Bargmann group. (This is shown very nicely in QUANTIZATION ON A LIE GROUP: HIGHER-ORDER POLARIZATIONS by V. Aldaya, J. Guerrero and G. Marmo).
+<--
+--> SO(3) -> SU(2)#
+The standard spatial rotation group $SO(3)$ needs to be extended by $\mathbb{Z}_2$, which yields $SU(2)$, because otherwise we are not able to describe spin $\frac{1}{2}$ particles.
+<--
+--> Mickelsson-Faddeev algebra#
+The algebra of fermionic non-Abelian charge densitites needs to be extended to the Mickelsson-Faddeev algebra (See [[http://physics.stackexchange.com/a/76653/37286|this answer]])
+<--
+<tabbox FAQ>
+<tabbox History>
+</tabbox>

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