Timeline for Does the classification of reductive groups follow from that of semisimple groups?
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| Dec 28, 2019 at 20:44 | comment | added | LSpice | Oh, right! I missed that we are considering the fundamental group abstractly, not with a specified embedding in the weight-modulo-root group, so that we can't distinguish the three copies of $\operatorname C_2$ inside $\operatorname W(\mathsf D_{2n})$, even though two of them really do give non-isomorphic groups in the non-triality case. Thanks! | |
| Dec 28, 2019 at 20:36 | comment | added | Jim Humphreys | @LSpice: Sorry for the delay in responding, but I had to search awhile to find older printings of my book GTM 21. In Theorem 32.1, not having yet embraced the idea of "root datum", I gave too superficial an account of isomorphism between "simple" algebraic groups. In the revised printing I added the obvious exception, the group of type D$_\ell$ with even rank $\ell\geq 6$. (What is actually proved here is the equivalent Theorem' following Chevalley which involves extending a map. A complicated but concrete method compared to Takeuchi, later Steinberg.) | |
| Dec 23, 2019 at 23:32 | comment | added | LSpice | @JimHumphreys, your answer is very nice, but it's not clear to me how it indicates that a connected simple group over an algebraically closed field "isn't always determined up to isomorphism by its root system and fundamental group". (Of course isogenies are relevant, but change the fundamental group, at least infinitesimally.) Could you say more? | |
| Dec 15, 2016 at 15:30 | vote | accept | D_S | ||
| Dec 14, 2016 at 21:12 | answer | added | Jim Humphreys | timeline score: 9 | |
| Dec 14, 2016 at 15:41 | comment | added | Jim Humphreys | @D_S: As nfdc23 indicates, the answer to your basic question is 'yes', but already over an algebraically closed field it's a bit tricky to state a comprehensive theorem. For example, even when $G$ is connected and simple (as an algebraic group), it isn't always determined up to isomorphism by its root system and fundamental group (as you assert and as I loosely formulated it in the first printing of my 1975 book). I'll try to write a detailed answer if I have time later today, including reductive groups in the mix. | |
| Dec 13, 2016 at 1:56 | comment | added | nfdc23 | Yes. For any field $k$ and connected reductive $k$-group $G$ there is the canonical isomorphism $(Z \times G')/\mu \simeq G$ where $G'$ is the connected semisimple derived group, $Z$ is the maximal central torus, and $\mu = Z \cap G'$ is a central $k$-subgroup scheme of $G'$ (and if $H$ is a connected semisimple $k$-group, $\mu \subset H$ is a central $k$-subgroup scheme, and $T$ is a $k$-torus equipped with an inclusion of $\mu$ then $G :=(T \times G')/\mu$ is connected reductive with derived group $G'$, maximal central torus $T$, etc.) | |
| Dec 13, 2016 at 0:05 | history | asked | D_S | CC BY-SA 3.0 |