Central extension
For any
Lie group G, if there exists a Lie group G' and a
surjective homomorphism with an
Abelian Lie group as its
kernel, such that there does not exist any right inverse (i.e. a homomorphism such that αβ is the
identity morphism), then we say G' is a
central extension of G.
 The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by E, P_{i}, C_{i} and L_{ij} (antisymmetric tensor) subject to








We can now give it a central extension into the Lie algebra spanned by E', P'
_{i}, C'
_{i}, L'
_{ij} (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the
center, that's why it's called a central extension) and








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