advanced_tools:group_theory:central_extension
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advanced_tools:group_theory:central_extension [2017/12/17 12:07] jakobadmin [Student] |
advanced_tools:group_theory:central_extension [2017/12/17 12:12] jakobadmin |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Central extensions are a standard trick to convert [[group_theory:notions:projective_representation|projective representations]] of some group into true representations of another group. | + | 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. | 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. | ||
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| * See page 178 in Moonshine beyond the Monster by Terry Gannon | * See page 178 in Moonshine beyond the Monster by Terry Gannon | ||
| - | **Important Examples:** | ||
| - | * 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). | ||
| - | * 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. | ||
| - | * 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]]) | ||
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| <tabbox Examples> | <tabbox Examples> | ||
| - | --> Example1# | ||
| + | --> 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). | ||
| <-- | <-- | ||
| - | --> Example2:# | + | --> 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]]) | ||
| <-- | <-- | ||
advanced_tools/group_theory/central_extension.txt · Last modified: 2017/12/17 12:26 by jakobadmin
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