Timeline for Is it true that $\mathfrak{g}=\mathfrak{g}_e\oplus[x,\mathfrak{g}]$?
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10 events
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| Feb 29 at 12:20 | comment | added | Hugo MTV | @SHP, in the case of Slodowy slices, the sum is not necessarily direct. In Gan and Ginzburg article transversality is to be understood as in diffrential geometry, that is why only surjectivity of the differential is mentionned. It is always direct at x=f by usual $\mathfrak{sl}_2$ theory. Yet, any Slodowy slice contains regular elements (as $G\cdot( f+\mathfrak{g}^e)$ is dense in $\mathfrak{g}$ and the set of regular elements is open). Hence by a dimension argument, if $e$ is not regular, the sum cannot be direct at the regular elements of the slice. | |
| May 18, 2018 at 13:51 | comment | added | SHP | @მამუკაჯიბლაძე It is stated, for example, in Gan--Ginzburg Quantization of Slodowy slices, Section 2.2. But I think it goes back to Slodowy's work. | |
| May 18, 2018 at 13:23 | comment | added | მამუკა ჯიბლაძე | @SHP What confuses me is that usually one formulates this property of slices only for $x=f$. Is it stated in this form somewhere? | |
| May 18, 2018 at 8:21 | comment | added | SHP | This is one of the main reasons why Slodowy slices are studied. They are transverse to the adjoint orbits, meaning that if $x\in f+\mathfrak{g}_e$, then $T_x\mathfrak{g}=T_x(f+\mathfrak{g}_e)\oplus T_x(G\cdot x)$ where $G$ is the adjoint group of $\mathfrak{g}$. Under the identification $T_x\mathfrak{g}=\mathfrak{g}$, we recover the identity in question. | |
| May 17, 2018 at 15:08 | answer | added | S. carmeli | timeline score: 1 | |
| May 17, 2018 at 11:38 | comment | added | მამუკა ჯიბლაძე | @YCor Oh yes sorry | |
| May 17, 2018 at 11:37 | comment | added | YCor | @მამუკაჯიბლაძე No: $e\notin [f,\mathfrak{g}]$ [$:=\mathrm{Im}(ad(f))$] in $\mathfrak{sl}_2$. | |
| May 17, 2018 at 11:35 | comment | added | მამუკა ჯიბლაძე | Something is wrong here: $e\in[x,\mathfrak g]\cap\mathfrak g_e$ for e. g. $x=f$ | |
| May 17, 2018 at 8:17 | history | edited | Simon Parker | CC BY-SA 4.0 |
added 185 characters in body
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| May 17, 2018 at 8:02 | history | asked | Simon Parker | CC BY-SA 4.0 |