Timeline for Explicit $2$-cocycle for $2^{1+2n}_+$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23 at 14:34 | comment | added | Antoine | Now I see your point, thank you! | |
| Jul 23 at 12:14 | comment | added | Dave Benson | For the linear part, you can use the "carry digit" cocycle familiar from your youth. | |
| Jul 23 at 11:49 | comment | added | Dave Benson | Cocycles form an abelian group. Just add them. | |
| Jul 23 at 11:44 | comment | added | Antoine | Yes, there are $B: V\times V\to \mathbb{F}_p$ and $q: V\to \mathbb{F}_p$, and q is non trivial for $E=p^{1+2n}_{-}$ (p odd). In what sense can I add them? | |
| Jul 23 at 11:05 | comment | added | Dave Benson | For $p$ odd, you don't have a quadratic form. You have a symplectic form giving the commutators, and independently a linear form giving the $p$th powers. There's no connection between them. So just treat the two cocycles separately and add. | |
| Jul 23 at 10:54 | comment | added | Antoine | Btw, regarding $E=p^{1+2n}_{-}$, I was thinking that $\beta(v,w)=\frac{1}{2}([v,w]+q(v)+q(w))$ is the natural choice, but it fails to be a cocycle. Do you suggest another candidate? | |
| Jul 17 at 4:23 | comment | added | Antoine | now it is clear, thank you again! | |
| Jul 16 at 16:39 | vote | accept | Antoine | ||
| Jul 16 at 16:32 | comment | added | Dave Benson | I guess the point is that if you lift elements of $V$ to $E$ then $\beta(x,y)$ tells you how the lift of the product differs from the product of the lifts. So $\beta(x,y)-\beta(y,x)$ tells you the commutator of the lifts because $V$ is commutative. The formula for the commutator in terms of the square tells you this is $q(x-y)-q(x)+q(y)$. And as mentioned before, minus is plus. I hope that helps. | |
| Jul 16 at 16:28 | comment | added | Antoine | Thanks a lot! I know where the condition $\beta(x,x)=q(x)$ comes from, but the other condition is not evident to me, does it follow from the cocycle equation? | |
| Jul 16 at 15:30 | comment | added | Dave Benson | It might have made it clearer what was going on if I had written some of the plusses as minuses. But then people would complain that it makes no difference, so why do it? You can't win, you know. | |
| Jul 16 at 14:47 | history | answered | Dave Benson | CC BY-SA 4.0 |