Editing Galilean transformation (section)

== Central extension of the Galilean group ==
One may consider<ref>{{harvnb|Bargmann|1954}}</ref> a [[Lie algebra extension#Central|central extension]] of the Lie algebra of the Galilean group, spanned by {{math|''H''′, ''P''′<sub>''i''</sub>, ''C''′<sub>''i''</sub>, ''L''′<sub>''ij''</sub>}} and an operator ''M'': 
The so-called '''Bargmann algebra''' is obtained by imposing <math>[C'_i,P'_j]=i M\delta_{ij}</math>, such that {{math|''M''}} lies in the [[center (algebra)|center]], i.e. [[Commutative operation|commute]]s with all other operators.

In full, this algebra is given as
:<math>[H',P'_i]=0 \,\!</math>
:<math>[P'_i,P'_j]=0 \,\!</math>
:<math>[L'_{ij},H']=0 \,\!</math>
:<math>[C'_i,C'_j]=0 \,\!</math>
:<math>[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!</math>
:<math>[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!</math>
:<math>[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!</math>
:<math>[C'_i,H']=i P'_i \,\!</math>
and finally 
:<math>[C'_i,P'_j]=i M\delta_{ij} ~.</math>
where the new parameter <math>M</math> shows up. 
This extension and [[projective representation]]s that this enables is determined by its [[Group cohomology#Projective representations and group extensions|group cohomology]].