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S. Weinberg in his book "The quantum theory of fields" vol. I says in page 86 that the full Poincare algebra is not semi-simple but its central charges can be eliminated (as he showed in the book).

For every finite dimensional semi-simple Lie algebra $\mathfrak{g}$ over a field $\mathbb{K}$ one can show $H^2 (\mathfrak{g};\mathbb{K})=0$. But as Weinberg has mentioend the full Poincare algebra is not semi-simple.

I need a reference for this fact that the second cohomology of the full Poincare algebra is zero. I would really appreciate if someone could help me about it.

Qmechanic
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Mahtab
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1 Answers1

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A perhaps very illuminating and nice argument is given in the thesis of Nesta van der Schaaf: https://www.math.ru.nl/~landsman/Nesta.pdf . Please take a look at Section 5.2. You get a full proof for your question, and in addition, a construction of the universal cover of the Poincare group.

On the Other hand, I can only recommend this thesis, it is written wonderfully for taking care of aspects of Projective Representations.

ProphetX
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