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I am learning Lie group and Lie algebra. I saw in a YouTube video "Supersymmetry lecture 02" from OpenCourseWare (OCW) at University of Cambridge at 11:17 that

$SO(3,1)$ is locally $SU(2) \times SU(2)$.

What does locally mean here? Does it refer to the fact that the Lie algebra of $SO(3,1)$ is equivalent to two independent $SU(2)$?

I am new to the subject, and any help is highly appreciated!

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    $\begingroup$ Related: physics.stackexchange.com/q/28505 $\endgroup$
    – AlmostClueless
    Commented Dec 21, 2023 at 15:50
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    $\begingroup$ actually, the complexification of $SO(3,1)$ is locally $SU(2)\otimes SU(2)$. As a real form, $SO(3,1)$ is non-compact so all its unirrep are infinite-dimensional, whereas there are finite dimensional unirreps of $SU(2)\otimes SU(2)$ $\endgroup$
    – ZeroTheHero
    Commented Dec 21, 2023 at 16:16
  • $\begingroup$ Doesn’t “locally” here mean “in the neighborhood of any group element”? “Globally”, the entire group isn’t that direct product. $\endgroup$
    – Ghoster
    Commented Dec 21, 2023 at 21:29
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    $\begingroup$ Also related: physics.stackexchange.com/a/682536/70245 $\endgroup$
    – Buzz ♦
    Commented Dec 22, 2023 at 2:27

1 Answer 1

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  1. Two Lie groups $G,H$ are locally isomorphic iff their corresponding Lie algebras $\mathfrak{g},\mathfrak{h}$ are isomorphic $\mathfrak{g}\cong\mathfrak{h}$, cf. e.g. this Math.SE post.

  2. The YouTube video is strictly speaking wrong: The two Lie algebras $so(3,1;\mathbb{R})$ and $su(2)\oplus su(2)$ are not isomorphic. However, their complexifications$^1$ are isomorphic $$so(3,1;\mathbb{C})~\cong~sl(2,\mathbb{C})\oplus sl(2,\mathbb{C}),$$ cf. e.g. this Phys.SE post.

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$^1$ Notice how the lecturer around 13:30 introduces an explicit imaginary unit $i$ into the construction.

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