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While almost all accessible references indicate/demonstrate that group SO(4) = SU(2)⊗SU(2), I've come across two references that state the relationship as SO(4) = SU(2)⊕SU(2).

Is the latter equation referring to the Lie algebra? Or is it a typo?

Clarification appreciated.

iSeeker
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    Could you give a reference saying that $SO(4)=SU(2)\otimes SU(2)$? What is true is that $$(SU(2)\times SU(2))/{\Bbb Z_2}\cong SO(4).$$ The $\oplus$ is for vector spaces, i.e., for Lie algebras. – Dietrich Burde Mar 01 '19 at 12:21
  • @DietrichBurde Sure: https://arxiv.org/pdf/quant-ph/0608186v2.pdf . And if the + is for the Lie Algebras, that answers my question. Many thanks – iSeeker Mar 01 '19 at 12:24
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    Is this the tensor product considered as representations? For the other isomorphism see here. – Dietrich Burde Mar 01 '19 at 12:42
  • @DietrichBurde It could also just be physicists using different notation (I have seen them use $\otimes$ for direct product before at least). – Tobias Kildetoft Mar 01 '19 at 12:58
  • @TobiasKildetoft I see. But then, how to deal with the non-trivial kernel $\Bbb Z_2$? – Dietrich Burde Mar 01 '19 at 13:00
  • @DietrichBurde That I don't know. – Tobias Kildetoft Mar 01 '19 at 13:07
  • @Dietrich Burde I'd already found that MIT paper, but to be honest, it goes beyond my current competence :-) . Two other related sources that make sense of this are https://math.stackexchange.com/questions/3646/recovering-the-two-su2-matrices-from-so4-matrix?noredirect=1&lq=1 and https://math.stackexchange.com/questions/2934457/the-map-from-su2-times-su2-to-so4 (PS This question is from a physical chemist, not a mathematician) – iSeeker Mar 01 '19 at 13:08
  • @iSeeker Yes, you are right. I know these posts already, but they do not mention a tensor product between these Lie groups. So this might be indeed a notation used by physicists. – Dietrich Burde Mar 01 '19 at 13:26
  • @Dietrich Burde Thanks for your replies and interest. – iSeeker Mar 01 '19 at 13:49

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