- #1

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

Thanks.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter WWGD
- Start date

- #1

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

Thanks.

- #2

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

- #3

lavinia

Science Advisor

Gold Member

- 3,241

- 625

Every Lie group is a manifold - in fact a differentiable manifold. I dont think there are any other fields that are a Euclidean space.

- #4

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

- #5

- 22,089

- 3,296

Hint: every "Lie field" must also be a commutative Lie group. And the connected commutative Lie groups are exactly of the form ##S^1\times ... \times S^1\times \mathbb{R}^n##.

More generally, you could be interested in topological fields. It turns out that the only connected, locally compact fields are ##\mathbb{R}## and ##\mathbb{C}##, but this is a tad more difficult to prove than the Lie case.

More generally, you could be interested in topological fields. It turns out that the only connected, locally compact fields are ##\mathbb{R}## and ##\mathbb{C}##, but this is a tad more difficult to prove than the Lie case.

Last edited:

- #6

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

- #7

lavinia

Science Advisor

Gold Member

- 3,241

- 625

they are not commutativeDo you mean that the restriction of the Lie field to the additive (Abelian) group is a Lie group? And aren't ##S^3, S^7 ## also connected, commutative Lie groups?

- #8

WWGD

Science Advisor

Gold Member

- 5,461

- 3,780

Yes, I just remembered and edited, sorry.

- #9

lavinia

Science Advisor

Gold Member

- 3,241

- 625

also the octonians are not a group.Yes, I just remembered and edited, sorry.

Here is a link that has a proof that the only spheres that are Lie groups are the circle and the 3 sphere.

http://math.stackexchange.com/quest...y-way-to-show-which-spheres-can-be-lie-groups

Share:

- Replies
- 5

- Views
- 5K