3 ms·

Technically the unit quaterions are not Spin(3), but only isomorphic to it, they are properly Sp(1) = GL(1, H). It's all fuzzy because the low dimensional class

Technically the unit quaterions are not Spin(3), but only isomorphic to it, they are properly Sp(1) = GL(1, H). It's all fuzzy because the low dimensional classical groups are all isomorphic to each other: SU(2) ~ Sp(1) ~ Spin(3).

gspr 5y ago
Indeed.
jacobolus 5y ago
This seems extraordinarily nitpicky. Like saying that unit complex numbers are technically not the group of plane rotations about a fixed point, but only isomorphic to it. Or for that matter like saying that the "real number line" is technically not a line, but only isomorphic to one.
- rnhmjoj 5y ago
  Well yeah, it is. The point I wanted to make is that these isomorphism are "exceptional"[1] and only hold for the lower dimensional groups. The general Spin group and quaterions are very different objects. [1]: https://en.wikipedia.org/wiki/Exceptional_isomorphism https://en.wikipedia.org/wiki/Exceptional_isomorphism
joppy 5y ago
That really depends on how you define Spin(3), for example some would define it as the compact simply-connected Lie group of a certain type, at which point the unit quaternions model of Spin(3) is as good as any other.
tobinfricke 5y ago
> Technically X is not Y, but only isomorphic to it Not sure this is a useful argument. If two structures are isomorphic, there is no way to tell them apart. If you can't tell them apart - maybe they are the same thing.
- nick__m 5y ago
  Isomorphism implies that there is a complete reversible mapping (a bijection) between structure A and B . Real number under addition are isomorphic¹ to positive real number under multiplication. Yet they are not the same objects. 1- The mapping is f(x)=e^x. See https://math.stackexchange.com/questions/573794/prove-that-mathbbr-and-the-interval-0-infty-have-the-same-cardinality https://math.stackexchange.com/questions/573794/prove-that-m... for the proof
  - Tainnor 5y ago
    When it comes to group theory, they are the same object. When we talk about things like rotation groups, we're not usually concerned with the way they are represented (much as we don't usually how the real or the complex numbers are constructed, for both of which there exist multiple different constructions).

gspr 5y ago

Indeed.

jacobolus 5y ago

This seems extraordinarily nitpicky. Like saying that unit complex numbers are technically not the group of plane rotations about a fixed point, but only isomorphic to it. Or for that matter like saying that the "real number line" is technically not a line, but only isomorphic to one.

rnhmjoj 5y ago
Well yeah, it is. The point I wanted to make is that these isomorphism are "exceptional"[1] and only hold for the lower dimensional groups. The general Spin group and quaterions are very different objects. [1]: https://en.wikipedia.org/wiki/Exceptional_isomorphism https://en.wikipedia.org/wiki/Exceptional_isomorphism

joppy 5y ago

That really depends on how you define Spin(3), for example some would define it as the compact simply-connected Lie group of a certain type, at which point the unit quaternions model of Spin(3) is as good as any other.

tobinfricke 5y ago

> Technically X is not Y, but only isomorphic to it Not sure this is a useful argument. If two structures are isomorphic, there is no way to tell them apart. If you can't tell them apart - maybe they are the same thing.

nick__m 5y ago
Isomorphism implies that there is a complete reversible mapping (a bijection) between structure A and B . Real number under addition are isomorphic¹ to positive real number under multiplication. Yet they are not the same objects. 1- The mapping is f(x)=e^x. See https://math.stackexchange.com/questions/573794/prove-that-mathbbr-and-the-interval-0-infty-have-the-same-cardinality https://math.stackexchange.com/questions/573794/prove-that-m... for the proof
- Tainnor 5y ago
  When it comes to group theory, they are the same object. When we talk about things like rotation groups, we're not usually concerned with the way they are represented (much as we don't usually how the real or the complex numbers are constructed, for both of which there exist multiple different constructions).