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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.

Examples

  1. 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, Pi, Ci and Lij (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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