# $SO(3,1)$ is locally $SU(2)\times SU(2)$, what does *locally* mean here?

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!

• Commented Dec 21, 2023 at 15:50
• 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 $SU(2)\otimes SU(2)$ Commented Dec 21, 2023 at 16:16
• Doesn’t “locally” here mean “in the neighborhood of any group element”? “Globally”, the entire group isn’t that direct product. Commented Dec 21, 2023 at 21:29
• Also related: physics.stackexchange.com/a/682536/70245
– Buzz
Commented Dec 22, 2023 at 2:27

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