Timeline for Isomorphism between Spin(3,2) and Sp(4, R)
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 17, 2013 at 15:40 | comment | added | paul garrett | @user36938 Aha! Thanks for the information! This provides some incentive to look at Galois actions on buildings... which I'd always neglected. "Barbs"! | |
| Jul 17, 2013 at 14:27 | comment | added | user36938 | @paul: Via algebraic-group techniques (Bruhat-Tits, cohomology, group schemes), the char-free classification over non-arch. local fields $k$ can be done with a dichotomy based on the parity of $n$. The cases $n \le 2$ have one cohomological invariant (and the relevant algebraic groups are not semisimple). If $n \ge 5$ then isotropicity holds for the quadratic space because absolutely simple semisimple $k$-groups not of type A must be isotropic (due to Bruhat-Tits theory). So by Witt cancellation, one is brought to $n=3, 4$. Exceptional isomorphisms yield two concrete cohomological invariants. | |
| Jul 17, 2013 at 12:54 | comment | added | paul garrett | @user36938, yes, of course, but I meant that the classification in char 2 is slightly more complicated, so while I myself know how to classify quadratic forms over local fields because I know the quaternion algebras, and such things, thus knowing that real orthogonal groups are classified by signature, and over p-adic fields we find that we need only classify up through dimension 4, etc., I myself do not know how to name/specify/distinguish isomorphism classes in char 2. | |
| Jul 17, 2013 at 3:04 | comment | added | user36938 | @paul: Arf invariants are irrelevant! For non-degenerate $(V,q)$ over any field $k$ with dim($V$) even, ${\rm{O}}(q)$ acts on the graded Clifford algebra of $(V,q)$, hence on the center of the "even" part. This center is finite etale of degree 2 since dim($V$) is even, so the center has automorphism scheme $\mathbf{Z}/(2)$. Hence, the action on that center defines ${\rm{O}}(q) \rightarrow \mathbf{Z}/(2)$ whose kernel is the char.-free definition of ${\rm{SO}}(q)$. For dim($V$) odd, the char.-free definition is ${\rm{SO}}(q) = {\rm{O}}(q) \cap {\rm{SL}}(V)$. | |
| Jul 16, 2013 at 16:38 | comment | added | paul garrett | Perhaps quadratic forms experts are less uneasy about Arf invariants than I am... :) | |
| Jul 16, 2013 at 10:28 | history | edited | user36938 | CC BY-SA 3.0 |
added 111 characters in body
|
| Jul 16, 2013 at 1:19 | comment | added | user36938 | @paul: even in char. 2 everything works perfectly well based on char-free definitions, as it must if there is to be a reasonable theory over $\mathbf{Z}$ for linking it up with Chevalley groups. The "correct" dichotomy for SO and Spin groups is not char. 2 versus char. $\ne 2$ but rather $\dim V$ being even or odd (i.e., B versus D). | |
| Jul 15, 2013 at 23:55 | comment | added | paul garrett | I do agree it's good to learn how to avoid "choice-of-coordinates" silliness. But/and some of the intrinsification is better understood after one has done things an ad-hoc way. (I think we all know this, although it is not high-rep to admit it.) Then there is the field-specific issue of "classification" of (non-degenerate) quadratic forms over a particular field, which is surely not solvable in general... But, over any algebraically closed field of char not 2 (such as complex...) it's just dimension, and a slightly non-trivial result: over $\mathbb R$, it's "signature". Not "just algebra". | |
| Jul 15, 2013 at 12:36 | comment | added | user36938 | @Jose: I am using standard notation among algebraists who study algebraic groups over an arbitrary field $k$, where non-degenerate quadratic forms are not classified by a "signature", but typically a lot more data. One writes ${\rm{SO}}(q)$ and ${\rm{Spin}}(q)$ for the connected semisimple $k$-groups associated to a non-degenerate quadratic space $(V,q)$, and when $q$ is split with $V$ of rank $n$ then they are denoted ${\rm{SO}}_n$ and ${\rm{Spin}}_n$. For $k=\mathbf{R}$, one writes ${\rm{SO}}(r,s)$ to denote ${\rm{SO}}(q)(k)$ for $q$ of signature $(r,s)$. So I prefer to leave it as is. | |
| Jul 15, 2013 at 10:20 | comment | added | José Figueroa-O'Farrill | To avoid confusion, I think you should perhaps write $\operatorname{SO}_{3,2}$ and $\operatorname{Spin}_{3,2}$ for $\operatorname{SO}_5$ and $\operatorname{Spin}_5$, respectively, and similarly, $\operatorname{Spin}_{3,3}$ for $\operatorname{Spin}_6$ in the last line. | |
| Jul 15, 2013 at 4:18 | history | edited | user36938 | CC BY-SA 3.0 |
added 118 characters in body
|
| Jul 14, 2013 at 21:55 | history | answered | user36938 | CC BY-SA 3.0 |