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I know you can take the General linear group of some vector space $V$: $GL(V)$.

For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic. I know that the three sphere admits a set of three linearly independent vector fields, the $\it{basis}$ of which can be represented by elements $SU(2)$ (or equivalently the $i,j,j$ of the quaternions).

So given such a space, how would I go about finding the General Linear group of that group (maybe that's a bad way to word it)?

Or maybe I should say how do I find the General linear group with that structure imposed upon it. How do I find out how that group differs from one on Euclidean 3-space, in a group theoretic manner. (apologies for poor terminology, my background is in physics).

I was hoping to do this for more general cases and groups like the special affine group of some particular space (for example).

R. Rankin
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    You need to be able to add elements in $G$ to define the product of matrices. You at least need a semiring. – Pedro Dec 23 '18 at 12:56
  • As an aside, what makes you think the 'special affine group' is isomorphic to the double cover of the Poincaré group? The dimensions don't seem to match. – Thomas Bakx Dec 23 '18 at 19:45
  • @ThomasBakx The special affine group: $$SA(4,R)=SL(4,R)\ltimes R^{1,3} $$ (for Minkowskian space) which is isomorphic to: $$SL(2,C)\ltimes R^{1,3}$$, which is the connected double cover of the Poincare group. I'm pretty sure that's right. I decided to ask about it here: https://physics.stackexchange.com/questions/450963/are-particles-always-represented-by-4-volume-preserving-transformations – R. Rankin Dec 30 '18 at 01:23
  • @ThomasBakx I could be wrong (though i'd like to know either way) https://math.stackexchange.com/questions/3048029/when-can-a-matrix-lambda-in-sl4-r-be-represented-by-sl2-c It would seem at the very least, The connected double cover is a subgroup of the SAG? – R. Rankin Dec 30 '18 at 01:30
  • @PedroTamaroff So wouldn't $SU(2)$ work just fine? Or if you prefer the terminology, the group $H^1$, the set of unit quaternions? – R. Rankin Dec 30 '18 at 01:33
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    @R.Rankin The identity matrix is special unitary, but twice the special matrix is not. Similarly, the sum of two unit quaternions is not, in general, a unit quaternion. – Pedro Dec 30 '18 at 02:01
  • Your question still makes no sense as written. – Moishe Kohan Dec 30 '18 at 02:17
  • @MoisheCohen Aloha, I reworded it, hopefully that helps somewhat – R. Rankin Dec 30 '18 at 02:34
  • @PedroTamaroff I must be a little confused, because I thought Dealing with the Lie Group, we have products being defined, not addition as in the Lie algebra. Anyway, I reworded the question, which hopefully makes it more clear. – R. Rankin Dec 30 '18 at 02:38
  • @R.Rankin What idea are you trying to capture? If you have a group and you want to talk about some kind of "invertible maps preserving the group structure" then that's just group automorphisms (or Lie group automorphism in the case of Lie groups). And I think Pedro means that in order to make sense of matrices with entries in the group (which you want to view as acting on linear combination of your "basis"), you need to have addition defined as well. – Pratyush Sarkar Dec 30 '18 at 05:22
  • @PratyushSarkar I was trying to find the group of continuous deformations/transformations of a space that preserve the volume of say a three-sphere (for example), I thought this approach would help me on the right track. I asked a very different approach here: https://math.stackexchange.com/questions/3023539/total-volume-preserving-transformations – R. Rankin Dec 30 '18 at 05:32
  • @R.Rankin In that case, I am not hundred percent sure but may be you want to look into divergence free vector fields and Killing fields?... – Pratyush Sarkar Dec 30 '18 at 05:46
  • @R.Rankin "I'm pretty sure that's right." - I'm pretty sure it's not. Where did you get it from in the first place? I've studied Wigner's work quite a lot and have never seen it. The fact that '4-volume preserving transformations' arise in particle representations is by no means a coincidence: the preservation of this 4-volume is precisely what is required to ensure a relativistically invariant theory (i.e. one where all inertial observers are equivalent). – Thomas Bakx Dec 30 '18 at 12:01
  • @ThomasBakx Thank you for the explanation, very helpful (: As for the other. thing: Special affine group is of course the Special linear version of the affine group, The double cover, I see expressed as above in various places, such as http://pages.cs.wisc.edu/~guild/symmetrycompsproject.pdf Please though, let me know how it's wrong so I can learn (: – R. Rankin Dec 31 '18 at 04:59

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