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I have a new relevant question. Let U, V, W be connected Lie groups, so that U is the semidirect product of V and W. By Schreier's theorem (Pontrjagin, Theorem 61), each group has a unique (up to an isomorphism) universal cover.

Question. If $U = V \rtimes W$, then is there a connection between $\tilde U, \tilde V, \tilde W$? If so, what would be the proof? Physicists assume that

$$\tilde U \simeq \tilde V \rtimes \tilde W$$ (see the case of Lorentz and Poincaré groups), so I am thinking & hoping a mathematician should know better.

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  • $\begingroup$ There is a comment here physics.stackexchange.com/questions/12341/… at the bottom of the page which says that topologically the direct product of two Lie groups is the same as its semidirect product. So that should clear my question. Can you prove it if it's true? $\endgroup$
    – DanielC
    Oct 29, 2017 at 22:59
  • $\begingroup$ I's interesting that nobody answered this yet. $\endgroup$
    – mma
    Mar 29, 2023 at 5:48

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