What does it mean for 2 groups to be locally isomorphic?

E.g. $SO(4)$ is locally isomorphic to $ SO(3)\times SO(3)$ -why not globally?

  • $\begingroup$ Where did you find this terminology? $\endgroup$ – Shaun Mar 9 at 0:33
  • $\begingroup$ Terry Tao's blog entry "Notes on local groups" defines the term, as well as other "localized" versions of group concepts. $\endgroup$ – FredH Mar 9 at 0:55
  • $\begingroup$ Based on Tao's blog, I would go with $G$ and $H$ are locally isomorphic if they contain local groups $G_0$ and $H_0$ which are isomorphic. $\endgroup$ – jgon Mar 9 at 1:32

The answer is: for groups it means nothing.

However, two topological groups $G,H$ are said to be locally isomorphic if there are open neighborhoods $V_G$, $V_H$ of $1_G$ and $1_H$ and a homeomorphism $f:V_G\to V_H$ such that for all $x,y\in V_G$, we have $xy\in V_G$ $\Leftrightarrow$ $f(xy)\in V_H$, and if so, then $f(xy)=f(x)f(y)$.

Classical theorems (see for instance Bourbaki, Lie groups and algebras, Chap 2) are that for Lie group over $\mathbf{R}$ or $\mathbf{Q}_p$ (or a product thereof, such as the solenoid $(\mathbf{R}\times\mathbf{Q}_p)/\mathbf{Z}[1/p]$), to be locally isomorphic is equivalent to having isomorphic Lie algebras.

Also, two complex Lie groups are locally isomorphic if there is $f$ as above that is, in addition, biholomorphic. This is also characterized as having isomorphic complex Lie algebras. It is stronger than being locally isomorphic as topological groups.


For Lie groups specifically what should happen is that two Lie groups are locally isomorphic iff they have isomorphic Lie algebras. $SO(4)$ and $SO(3) \times SO(3)$ are locally isomorphic because they have isomorphic Lie algebras

$$\mathfrak{so}(4) \cong \mathfrak{so}(3) \times \mathfrak{so}(3)$$

or, said another way, because the have isomorphic universal covers

$$\text{Spin}(4) \cong \text{Spin}(3) \times \text{Spin}(3)$$

(see spin group). This is an example of an exceptional isomorphism.


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