Timeline for Complete reducibility of representations of reductive algebraic groups
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8 events
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| Jan 28, 2016 at 19:17 | comment | added | zeno | To show that a reductive group is an almost-product of a torus and a semisimple group requires knowing that a semisimple group equals its derived group. This can be deduced from knowing that a semisimple group is an almost-product of simple groups. In characteristic zero, this last fact follows easily from the similar statement for semisimple Lie algebras, but in general it seems to require using roots. | |
| Jan 28, 2016 at 14:09 | comment | added | Jim Humphreys | The more difficult question is to get complete reducibility in char 0 for (say connected) reductive groups from the Borel-Tits definition. It seems to take a lot of work to show that such a group is the almost-direct product of a torus (for which all rational representations are completely reducible in any characteristic) and a semisimple group (possibly trivial). The root structure gets one close to the full classification. Maybe there's a shortcut, but in prime characteristic complete reducibility usually fails for semisimple groups. | |
| Jan 28, 2016 at 14:07 | comment | added | Jim Humphreys | As zeno points out, there are explicit statements. Actually, my early Chapter V on characteristic 0 theory does treat the case of a semisimple group with few preliminaries other than Weyl's complete reducibility theorem: see 13.2 and 14.3. [to be continued] | |
| Jan 27, 2016 at 19:20 | comment | added | zeno | jmilne.org/math/CourseNotes/iAG200.pdf Chapter 22, Section o. | |
| Jan 27, 2016 at 18:33 | comment | added | Jim Humphreys | @Jason: Presumably my British ancesters were mostly illiterate, so I'm used to all the variant spellings. Anyway, the question being asked is probably too elementary for this site. The concept of "reductive" algebraic group originates in the work of Borel-Tits, but related ideas in characteristic 0 are older in the theory of Lie groups. There is an identification (from the Chevalley classification) of semisimple Lie groups and semisimple algebraic groups over $\mathbb{C}$. Then the question reduces to "reductive" Lie algebras and Weyl's complete reducibility theorem. | |
| Jan 27, 2016 at 15:20 | comment | added | Jason Starr | You have misspelled the name of James Humphreys. | |
| Jan 27, 2016 at 14:26 | review | First posts | |||
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| Jan 27, 2016 at 14:23 | history | asked | Mario | CC BY-SA 3.0 |