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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!

Qmechanic
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1 Answers1

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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.

Qmechanic
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