# Tag Info

## Hot answers tagged lie-groups

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### A question about Lie groups

A Lie algebra $\frak g$ determines a simply connected Lie group. Let's call it $G$. It is the only simply connected Lie group with that Lie algebra. More precisely, if $H$ is a simply connected Lie ...
• 53.4k
13 votes

### Representation theory of $\mathrm{GL}_n(\mathbb{Z})$

For $n=2$, these groups are very close to free groups, so their representation theory is not very tractable. For $n \geq 3$, however, there is a beautiful story arising from Margulis superrigidity. ...
• 42.5k
6 votes

### Is $SO(N)$ a sphere?

Spheres are simply connected but $\operatorname{SO}(3)$ is not simply connected. Hence they cannot be homeomorphic, and a fortiori they cannot be isometric. If $a$ is a unit quaternion, i.e. a ...
• 11.4k
4 votes

### What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?

Any direct sum of a $n$-step nilpotent Lie algebra and a nonzero abelian Lie algebra satisfies the required property. Acknowledgement: After posting this answer, I realized that this class of examples ...
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4 votes
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• 10.7k
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### On Haar measure and Spherical measure

Yes, (of course, up to scalar multiples), the $U(n)$-invariant measure on $S^{2n-1}$ must be the same as the $O(2n)$-invariant measure on $S^{2n-1}$... for example, because the $U(n)$-invariant ...
• 22.3k
1 vote

### Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?

If you want to construct the multiplication table i suggest you lok at the explicitly definned product in the article https://www.cambridge.org/core/services/aop-cambridge-core/content/view/...

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