Timeline for When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 2, 2020 at 14:11 | comment | added | YCor | I finally asked $n=8$ as separate question mathoverflow.net/questions/359159 | |
| Sep 28, 2016 at 23:46 | comment | added | YCor | @Skip I think you could consider writing a separate answer for the case $n=8$ | |
| Sep 12, 2016 at 23:03 | comment | added | Skip | ... and moreover all 3 are $K$-defined because the Tits algebras of these three representations sum to zero in the Brauer group and you assume that 2 of them are already zero. Therefore, $G$ is Spin($\phi$) for an 8-dimensional quadratic form $\phi$ with trivial discriminant and trivial Clifford invariant, i.e., is an 8-dimensional form in $I^3$, so it is a scalar multiple of a 3-Pfister form. | |
| Sep 12, 2016 at 23:02 | comment | added | Skip | If $G$ is split simply connected of type $D_4$ then it has 3 inequivalent 8-dimensional irreducible representations that are interchanged by an automorphism of $G$ of order 3. The same is true if $G$ is Spin($q$) for a 3-Pfister quadratic form $q$ by Galois descent or using the explicit description of triality from the Book of Involutions. If $G$ is a $K$-form of one of these such that at least 2 of the 8-dimensional representations are $K$-defined, then $G$ has inner type $D_4$ (because outer types have at most 1) ... | |
| Sep 12, 2016 at 7:47 | comment | added | YCor | @Skip How do you prove this last statement (or what's a reference)? It's equivalent to the claim that some outer automorphism of order 3 of $\mathrm{Spin}(\phi)$ is defined over the ground field $K$. | |
| Sep 11, 2016 at 23:46 | comment | added | Skip | For $n = 8$, the answer is the same: if the special orthogonal groups of $\phi$ and $\psi$ are isogenous then they are isomorphic (done) or related by triality. More than 1 SO quotient of Spin($\phi$) is SO of a q.form iff $\phi$ is similar to a 3-Pfister quadratic form, in which case all three quotients are isomorphic to SO($\phi$). | |
| May 11, 2015 at 10:06 | vote | accept | jmc | ||
| Mar 24, 2015 at 20:23 | history | edited | YCor | CC BY-SA 3.0 |
gave the picture for $n=2$
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| Mar 24, 2015 at 19:56 | history | edited | YCor | CC BY-SA 3.0 |
improved to all $n\neq 8$
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| Mar 24, 2015 at 19:40 | history | answered | YCor | CC BY-SA 3.0 |