Timeline for Do Lie algebras of non-smooth group schemes over a field admit a Jordan decomposition?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| May 2 at 18:44 | vote | accept | LSpice | ||
| May 2 at 16:55 | answer | added | SeanC | timeline score: 1 | |
| Jan 1, 2023 at 11:38 | comment | added | LSpice | @hm2020, re$\DeclareMathOperator\Lie{Lie}$, obviously you are familiar with definitions of the Lie algebra and know where they are studied. Whereas many authors write $\Lie(G)$ for the Lie algebra, I write $\Lie(G)(k)$ for the very same Lie algebra, defined however you please (as you say, in terms of derivations is fine; or I like $\ker(G(k[\epsilon])) \to G(k))$), and $\operatorname{Lie}(G)$ for the resulting vector-group $A \mapsto \Lie(G)(k) \otimes_k A$. It is just different notation for the same object you know. | |
| Jan 1, 2023 at 9:45 | comment | added | hm2020 | what is the precise definition of $\mathfrak{g}(k)$ and where is this studied? If $G$ is a group scheme some authors define $Lie(G)$ using left invariant derivations of the Hopf algebra $A$ (over the base ring $k$). Some authors define it as $Der_k(A,\kappa(e))$ where $e$ is the identity. What is $Lie(G)(k)$ in these situations? | |
| Dec 31, 2022 at 16:30 | comment | added | LSpice | @hm2020, re, $\operatorname{Lie}(G)$ is, or can be treated as, a group scheme like any other (in this case a vector group scheme); e.g., if $G = \operatorname{GL}_n$, then $\mathfrak g = \mathfrak{gl}_n$, so $\mathfrak g(k) = \mathfrak{gl}_n(k)$. Many people use $\mathfrak g$ for what I call $\mathfrak g(k)$, and use some other decoration for the vector group scheme. If it bothers you and you believe it's so abnormal as to be likely to confuse others, then feel free to edit. | |
| Dec 31, 2022 at 16:23 | comment | added | hm2020 | For an affine $k$-group scheme $G:=Spec(A)$ we may define $G(k)$ to be the $k$-points of the affine scheme $G$. If $\mathfrak{g}:=Lie(G)$ is the Lie algebra of $G$ - what do you mean when you write $\mathfrak{g}(k)$? | |
| Dec 31, 2022 at 0:07 | history | edited | LSpice | CC BY-SA 4.0 |
Typos
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| Dec 31, 2022 at 0:02 | history | asked | LSpice | CC BY-SA 4.0 |