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In Valter Moretti's book (Spectral Theory and Quantum Mechanics), page 578, it is said that the Poincaré group is semisimple, but Wikipedia says otherwise.

Moretti mentions it in order to ensure that the Poincaré group satisfies the assumptions of the Bargmann theorem.

Is there something I am missing? Is Moretti committing a mistake? Are there several Poincaré groups, only some of which are semisimple? Do they all have vanishing degree $2$ cohomology (the essential condition in Barmann's theorem)?

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    $\begingroup$ @ Valter Moretti On page 742 of the current edition you/he say "Physically important cases where Bargmann’s theorem applies are [Bar54] SL(2,C) (the universal covering of the proper orthochronous Lorentz group) and the universal covering of the connected component at the identity of the Poincaré group. In particular the Lie algebra of the former is semisimple. " I read this to mean that only the Lorentz group is semisimple, So I think it is already corrected :) $\endgroup$
    – mike stone
    Feb 4, 2021 at 18:53
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    $\begingroup$ Thanks Mike, indeed it seemed to me that I already spotted that erroneous statement when preparing the second edition. $\endgroup$ Feb 4, 2021 at 18:57

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Yes, indeed that statement has been already corrected in the second edition of my book. Poincare group is a semi direct product of $SL(2,C)$ and the abelian Lie group $R^4$. The former is a (semi) simple Lie group, this is enough for extending the thesis of Bargmann’s theorem. The proof appears already in Bargmann’s paper.

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    $\begingroup$ Waow, I didn't expect to get an answer from the author of the book! By the way, dear Mr Moretti, I would like to warmly thank you for your book; I haven't found any other reference having the level of mathematical rigor I need together with such an insightful content. I recommend it to every mathematician interested in quantum mechanics! $\endgroup$
    – Plop
    Feb 4, 2021 at 20:07
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    $\begingroup$ Many thanks. But it is not true. Even the second edition contains some errors :(:(. However in around 800 pages it is impossible not to insert mistakes of various nature...I hope the third edition (if any) will be more correct. $\endgroup$ Feb 4, 2021 at 20:13

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