You kind of have it the other way around. If we can endow ##\mathbb R^n## with some "sufficiently nice" multiplication operation, then ##S^{n-1}## will be a group. But the converse isn't clear - why can't we have a group operation on ##S^{n-1}## that doesn't necessarily extend to a nice multiplication on ##\mathbb R^n##?homeomorphic said:Most spheres are not Lie groups. You could always turn the underlying set into a group, but then there's no point in it being a sphere.
I think this has something to do with division algebras, maybe. I think if it were a Lie group, maybe you could take the group algebra and get a division algebra. Division algebras only exist in certain dimensions: 1, 2, 4, and 8. So you get spheres of dimension 1 less: 1, 3, and 7. And 7 doesn't work because the octonians, the dimension 8 division algebra, are non-associative, so you can't get a group by taking the unit octonians.
So the only spheres that are groups are S^1 and S^3.
Don't quote me on this, though, since I haven't quite thought it through. Just came up with it off the top of my head. I think it's right, though.
The end result is correct, anyway:
http://en.wikipedia.org/wiki/3-sphere
You kind of have it the other way around. If you can endow Rn with some nice "multiplication" operation, then Sn−1 will be a nice group. But the converse isn't clear - why can't we have a nice group operation on Sn−1 that doesn't necessarily extend to a nice multiplication on Rn?
Yes, that result is true: S^1 and S^3 (and S^0) are the only spheres that are Lie groups.homeomorphic said:It isn't clear, but according to wikipedia and probably some other sources I remember, it is true that only S^1 and S^3 are Lie groups. So, my speculation was that given a group operation on S^n-1, you can construct a division algebra of dimension n somehow. But it's not clear how. The wikipedia article also seems to indicate that the proof should proceed along these lines.
Most proofs I know don't mention real division algebras; it typically goes the other way around: once you have this result, you get a restriction on the possible division algebras.
A hyper-sphere, also known as an n-sphere, is a generalization of a sphere to higher dimensions. It is a geometric shape in n-dimensional space that is defined as the set of all points that are a fixed distance from a central point.
A hyper-sphere differs from a regular sphere in that it exists in n-dimensional space instead of just three-dimensional space. This means that it has n+1 dimensions, while a regular sphere only has three dimensions (length, width, and height).
The properties of a hyper-sphere include its radius, diameter, volume, and surface area. These properties can be calculated using formulas specific to n-dimensional space.
Hyper-spheres have many applications in mathematics and science, particularly in geometry, topology, and physics. They are used to model higher-dimensional spaces, such as in string theory, and in studying the behavior of particles in n-dimensional space.
While it is difficult to imagine or visualize a hyper-sphere in higher dimensions, there are some real-world examples that can help us understand the concept. For instance, a 3-sphere can be thought of as a four-dimensional ball, and a 4-sphere can be thought of as a five-dimensional hypersphere. However, it should be noted that these are just analogies and not actual hyper-spheres in our three-dimensional world.