Timeline for What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2015 at 17:03 | vote | accept | Gest2015 | ||
| Oct 10, 2015 at 13:41 | answer | added | Mikhail Borovoi | timeline score: 7 | |
| Oct 9, 2015 at 20:33 | comment | added | Mikhail Borovoi | I agree with Matthias Wendt. I will post an answer tomorrow. | |
| Oct 9, 2015 at 18:10 | comment | added | Gest2015 | I mean it is the algebraic fundamental group as defined by Borovoi in "Abelian Galois cohomology of reductive groups" | |
| Oct 9, 2015 at 16:22 | review | Close votes | |||
| Oct 10, 2015 at 23:10 | |||||
| Oct 9, 2015 at 14:13 | comment | added | Jim Humphreys | @Gest2015: Can you be explicit about "in the sense of Borovoi"? Presumably all definitions are equivalent over $\mathbb{C}$ (which I should have written instead of "in characteristic 0" at the end of my comment). | |
| Oct 9, 2015 at 14:11 | comment | added | Matthias Wendt | I think it follows from Example 1.6 of Borovoi's "Abelian Galois cohomology of reductive groups" that the algebraic fundamental group is, independently of the base field, given as in the comment of Igor Rivin. | |
| S Oct 9, 2015 at 13:51 | history | suggested | Z.A.Z.Z | CC BY-SA 3.0 |
correct the title
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| Oct 9, 2015 at 13:51 | review | Suggested edits | |||
| S Oct 9, 2015 at 13:51 | |||||
| Oct 9, 2015 at 13:14 | comment | added | Gest2015 | In fact, wat I mean is the algebraic fundamental group of $SO_r(k)$ (resp SP_{2r},...) in the sens of Borovoi. | |
| Oct 9, 2015 at 12:31 | comment | added | Jim Humphreys | It's important to make explicit what you mean by "fundamental group" in this context. Beyond the usual notion in algebraic topology, there is an algebraic notion introduced by Chevalley in the study of semisimple (or reductive) algebraic groups over an arbitrary algebraically closed field: the quotient of the abstract group of weights of a maximal torus by the actual character group of the torus. (This turns out to be equivalent to the topological definition in characteristic 0.) | |
| Oct 9, 2015 at 11:38 | comment | added | YCor | Does the question makes sense? the fundamental group is defined for an algebraic group, not for the group of $k$-points of an algebraic $k$-group. | |
| Oct 9, 2015 at 9:09 | history | edited | Gest2015 | CC BY-SA 3.0 |
added 2 characters in body
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| Oct 9, 2015 at 9:07 | comment | added | Gest2015 | Yes I know that, but I need to know it over an algebraicaly closed field of characteristic zero, not necessarily $\mathbb C$ | |
| Oct 9, 2015 at 8:53 | comment | added | Igor Rivin | Are you thinking of these as algebraic groups? Over $\mathbb{C}$ the fundamental group of $SO(n)$ is $\mathbb{Z}/2 \mathbb{Z}$ for $n > 2,$ $\mathbb{Z}$ when $n=2,$ while the symplectic group is simply connected. | |
| S Oct 9, 2015 at 8:01 | history | suggested | Z.A.Z.Z | CC BY-SA 3.0 |
correct some mistakes
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| Oct 9, 2015 at 8:00 | review | Suggested edits | |||
| S Oct 9, 2015 at 8:01 | |||||
| Oct 9, 2015 at 7:55 | history | asked | Gest2015 | CC BY-SA 3.0 |