Timeline for Emergence of the orthogonal group
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 22, 2020 at 2:05 | comment | added | Francois Ziegler | Perhaps the next person to consider (and famously parametrize, again without naming them) “les coëfficients propres à effectuer la transformation de deux systèmes de coordonnées rectangulaires” is Cayley (1846, equation 16 on p. 120). | |
| Jun 19, 2020 at 16:18 | vote | accept | Francois Ziegler | ||
| Jun 19, 2020 at 14:36 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a remark about Euler's parametrization of the orthogonal group in dimension 4.
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| Jun 19, 2020 at 13:38 | comment | added | Robert Bryant | @FrancoisZiegler: Yes, of course, the formula $(5)$ what I meant. I assumed from the beginning that Euler was doing that because he essentially says in the text that he wants to show how to parametrize the rational elements of $\mathrm{O}(4)$, i.e., solve the equations defining a matrix in $\mathrm{O}(4)$ in rational numbers.. (Although I confess that my Latin is very bad, I am sure that this is essentially what he is saying.) | |
| Jun 19, 2020 at 13:27 | comment | added | Francois Ziegler | When you say “the product” you mean Euler’s in (5), right? That’s brilliant and would show that he was not 1 but 2 steps ahead of everyone... One has to wonder if Gauss had read this paper; maybe even Cayley / Hamilton absorbed more of it than they let on. | |
| Jun 19, 2020 at 12:52 | comment | added | Robert Bryant | @FrancoisZiegler: I think that the map $c = \mathrm{diag}(1,-1,-1,-1)$ (which is conjugation in the quaternions) is there so that the product will parametrize $\mathrm{O}(4)$, i.e., the product is the matrix for $$X\mapsto Ac(BX) = A\overline{BX} = A\overline {X}\,\overline{B}.$$ This shows how Euler parametrizes the non-identity component of $\mathrm{O}(4)$ via left multiplication by two (unit) quaternions $A$ and $B$ and conjugation as $AcB$. To get the identity component, i.e., $\mathrm{SO}(4)$, you'd use the product $AcBc$, for the linear map $X\mapsto Ac(B(cX)) = AX\overline{B}$. | |
| Jun 19, 2020 at 11:50 | comment | added | Francois Ziegler | About $\mathrm O(5)$: absolutely and again I hadn’t read that far. Euler seemed only limited by the length of the alphabet :-) If you would include that in the answer I’d like to accept it. About quaternion multiplication, I think he almost had it but inserted a $$\mathrm{diag}(1,-1,-1,-1)$$ for which I’d love to understand the reason — see formula (5) of this answer which goes into more details. | |
| Jun 19, 2020 at 11:02 | comment | added | Robert Bryant | @FrancoisZiegler: Thanks for pointing out Euler's 1771 paper. I had a look at it. Doesn't he consider $\mathrm{O}(5)$ as well, starting in $\S28$? (Of couse, he didn't name it as a group.) Also, he has the formula for the double cover $S^3\to\mathrm{SO}(3)$ in $\S33$ and quaternion multiplication in $\S34$! I had not known this before. | |
| Jun 18, 2020 at 19:18 | comment | added | Francois Ziegler | Right, that’s how we recognize what group Lie is talking about. (I’d say he used $p_k$ for the hamiltonian generating infinitesimal translations in the $k$th direction.) I also recall now that Euler could be said to have “considered” — but certainly not named — $\mathrm O(4)$ in (1771, §20). But that was rather isolated and ahead of his time. | |
| Jun 18, 2020 at 14:49 | comment | added | Robert Bryant | @FrancoisZiegler: Hmmm. I remember that Lie used $p_k$ for what we call the vector field $\frac{\partial}{\partial x_k}$, so he's discussing the group acting on $\mathbb{R}^n$ with 'infinitesimal generators' $x_\nu\,\frac{\partial}{\partial x_k} -x_k\,\frac{\partial}{\partial x_\nu}$, which is, of course, what we now call the (special) orthogonal group acting on $\mathbb{R}^n$. He surely knew that this was a linear action on $\mathbb{R}^n$. Lie's whole point was that the infinitesimal generators determine the group, so specifying the group by its Lie algebra would have been natural for him. | |
| Jun 18, 2020 at 14:11 | comment | added | Francois Ziegler | I don’t know! The problem with Lie and Killing is that they are hard to catch talking about anything but Lie algebras. E.g. Lie (1893, p. 317) describing $\mathfrak{so}(n)$ as “the group $x_\nu p_k-x_k p_\nu\quad (\nu,k=1\dots n)$”. | |
| Jun 18, 2020 at 13:27 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Minor edits
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| Jun 18, 2020 at 13:24 | comment | added | Robert Bryant | @FrancoisZiegler: I wouldn't venture to say that. From what Carlo wrote in his answer, it would appear that Hurwitz already was dealing with $\mathrm{SO}(n)$ and $\mathrm{U}(n)$ as matrix groups in 1897, which seems pretty advanced for the time. Are you sure that Lie and/or Killing didn't recognize that the stabilizer groups of nondegenerate quadratic forms were of Lie's type B or D? Since the relation between 'linear fractional transformations' and $\mathrm{SL}(2)$ was well understood by Lie, you would expect that he'd have realized that 'projective groups' were matrix groups in disguise. | |
| Jun 18, 2020 at 12:55 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a remark about the term 'orthogonal group' in Cartan
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| Jun 18, 2020 at 12:21 | comment | added | Francois Ziegler | Thanks! I should definitely have read further. Would you say that Cartan has a chance of being first to “consider” the group, though not name it other than by one subscripted letter? Ironically Frobenius (1878) seems to do almost the opposite... | |
| Jun 18, 2020 at 9:26 | history | edited | Robert Bryant | CC BY-SA 4.0 |
added 213 characters in body
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| Jun 18, 2020 at 9:19 | history | edited | Robert Bryant | CC BY-SA 4.0 |
added 213 characters in body
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| Jun 18, 2020 at 9:12 | history | answered | Robert Bryant | CC BY-SA 4.0 |