Timeline for Are two distinct Weyl chambers always disjoint?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 14, 2016 at 6:27 | comment | added | Lior Silberman | In paragraph 1 we show that $H$ and $MA$ have the same connected component (as asserted in paragraph 2). Then in paragraph 3 we show (in the algebraic category) that $MA$ is connected. It follows that $MA$ is the connected component of $H$, which is a normal subgroup of $H$. | |
| Oct 9, 2016 at 18:14 | comment | added | Ilia Smilga | Sorry for coming back to this two years later. If you are still around, could you please explain to me the logical step at the beginning of the third paragraph? You know that $A$ is characteristic in $MA$, and you want to show it is normal in $H$. Let $h \in H$; then if $h(MA)h^{-1} = MA$, certainly this implies that $hAh^{-1} = A$. But how do you obtain the first equality? | |
| Apr 16, 2014 at 2:41 | comment | added | Lior Silberman | Because "real semisimple Lie group" can mean one of two things. It can mean the group of $\mathbb{R}$-points of a semisimple linear algebraic group defined over $\mathbb{R}$, or it can mean a Lie group whose Lie algebra is a direct sum of simple Lie algebras. | |
| Apr 15, 2014 at 14:37 | comment | added | Ilia Smilga | At least one point does not seem clear to me. Why do you treat separately the algebraic and the smooth category? The equality $H = Z_G(A)$ is either true or false, regardless of the category we are working in; why would you need to show it twice? | |
| Apr 3, 2014 at 18:36 | comment | added | Lior Silberman | I added a reference. | |
| Apr 3, 2014 at 18:35 | history | edited | Lior Silberman | CC BY-SA 3.0 |
reference added
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| Apr 3, 2014 at 18:18 | history | edited | Lior Silberman | CC BY-SA 3.0 |
Fixed erroneous argument
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| Apr 3, 2014 at 12:44 | comment | added | Jim Humphreys | Concerning your remark toward the end: In the algebraic setting, it isn't true in all cases that the centralizer of a semisimple element is connected. Can you clarify? (And I guess a reference is really needed here.) | |
| Apr 3, 2014 at 10:23 | history | answered | Lior Silberman | CC BY-SA 3.0 |