Timeline for From Weyl groups to Weyl groupoids?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 1, 2016 at 15:58 | comment | added | Ben Webster♦ | @SaalHardali You can change what the accepted answer is. | |
| May 1, 2016 at 12:18 | comment | added | Saal Hardali | @user89334 Thanks. Silly me, I do understand now why my construction is wrong. However, as a non expert on this topic I'm pretty confused by some of the wrong remarks on this thread. I feel like there ought to be at least one satisfying invariant geometric way to incorparate all the interrelated information contained in the borel algebras cartan algebras and weyl group. I'm sorry to have accepted this answer on the false premise that I had such a construction. Your edited part might be what i'm looking for but i still need to think that through. Thanks again! | |
| May 1, 2016 at 8:57 | comment | added | Uri Bader | @Saal, instead of me trying to convience you there isn't, can you try to write down one? Incidentally the last sentence in my previous comment is false as stated. also I earased a false comment I made here. There is a correct conceptual reason for which I wrote these wrong statements, and I will edit it into my answer when I'll find the time. | |
| May 1, 2016 at 6:50 | comment | added | Saal Hardali | @user89334 Thanks! I still don't really understand why there's no functor from $\mathcal{B} \to \mathcal{W}$ | |
| May 1, 2016 at 6:37 | comment | added | Uri Bader | @Saal Hardali that could be a very good reason indeed to accept the answer, only it is not correct and had not claimed by anyone. In fact there is no such canonical functor as you describe. On the other hand there are $W$ many functors $\mathcal{W}\to\mathcal{B}$ on which the abstract Weyl group I described in my answer acts. These functors could be thought of as choices of Weyl chambers in the abstract Cartan. | |
| May 1, 2016 at 5:45 | comment | added | Saal Hardali | @BenWebster I accepted yours since it really gave me the last piece I was missing. I have the impression that the true definition is in some sense the one S. Carnahan eluded to, namely, let the groupoid in my question be called $\mathcal{B}$ and let $\mathcal{W}$ be the correct weyl groupoid with objetcs cartan subalgebras and morphisms adjoint actions of $\mathfrak{g}$. There's a functor $\mathcal{B} \to \mathcal{W}$ which takes a borel to its abstract cartan. The automorphism group of that functor is the weyl group. | |
| May 1, 2016 at 5:22 | comment | added | Uri Bader | In any case it seems that my impression, that it was all about making the definitions choice free, wasn't what the OP wanted, and what you wote satisfied him. | |
| May 1, 2016 at 2:56 | comment | added | Ben Webster♦ | @user89334 Fair enough. Like I said, I wasn't really sure what the user wanted. | |
| Apr 30, 2016 at 20:05 | comment | added | Uri Bader | I must say that I fail to see how any of this is choice free. Surely, everything here is independent of the choices made, but choices are made. | |
| Apr 30, 2016 at 19:29 | vote | accept | Saal Hardali | ||
| May 1, 2016 at 16:28 | |||||
| Apr 30, 2016 at 19:29 | comment | added | Saal Hardali | Thank you! That was helpful. Mainly what I was after is a way to remember what choice gives me what and in what sense are they equivalent. S. Carnahan gave me the idea of encoding it in a torsor $\mathcal{W} \to G/B \to G/T$ where a point in the base is a cartan subalgebra and a point in the total space is a borel subalgebra. The weyl group acts transitively on the fibers. Does this work? | |
| Apr 30, 2016 at 19:28 | comment | added | LSpice | As the sidebar informs, this point of view is also discussed in mathoverflow.net/questions/103751/… . | |
| Apr 30, 2016 at 19:17 | history | answered | Ben Webster♦ | CC BY-SA 3.0 |