Timeline for Question about coadjoint orbits of compact connected Lie groups
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2022 at 22:43 | vote | accept | Mira | ||
| Nov 9, 2022 at 14:42 | answer | added | Mikhail Borovoi | timeline score: 2 | |
| Nov 6, 2022 at 20:13 | comment | added | Mikhail Borovoi | For today: Write $\frak g=\frak z \oplus \frak s$, $\frak g^*=\frak z^* \oplus \frak s^*$, where $\frak z$ is the center of $\frak g$, and $\frak s=[\frak g,\frak g]$ is the derived Lie algebra. Then the natural projection $\frak g^* \to \frak s^*$ induces an isomorphism of the symplectic varieties $G\cdot r$ and $G^{\rm ad}\cdot r_{\frak s}$ and preserves the Killing form. Here $G^{\rm ad}=G/Z(G)$ , and $r_{\frak s}$ denotes the projection of $r$ to $\frak s^*$. Note that ${\frak s}={\rm Lie}\,G^{\rm ad}$ is a semisimple Lie algebra. | |
| Nov 6, 2022 at 20:12 | comment | added | Mikhail Borovoi | I will type an answer tomorrow or on Tuesday. | |
| Nov 6, 2022 at 19:20 | comment | added | Mira | @MikhailBorovoi, I'm sorry I didn't add these details in my post! In this formula $\hat{X}$ is a tangent vector of $T_\alpha(\mathcal{O}_r)$ , where $\hat{X}= \frac{d}{dt}\rvert_ {t=0} e^{-t X}\cdot\alpha$. | |
| Nov 6, 2022 at 19:00 | comment | added | Mikhail Borovoi | If you indeed want to get an answer, please try to explain your notation.... | |
| Nov 6, 2022 at 18:57 | comment | added | Mikhail Borovoi | The first formula $$\omega_\alpha(\hat{X},\hat{Y})= -\alpha([X,Y]), \alpha \in \mathfrak{g}^*, \quad X,Y, \in \mathfrak{g}.$$ is not clear. What is the relation between $X$ and $\hat X$? | |
| Nov 6, 2022 at 18:43 | comment | added | Mira | @MikhailBorovoi, $\hat{X}$ are $\hat{Y}$ are tangent vectors in $T_\lambda(\mathcal{O_r})$, which is identified with the set $\lbrace [Z,\lambda] , Z \in \mathfrak{g} \rbrace $. | |
| Nov 6, 2022 at 18:23 | comment | added | Mikhail Borovoi | Could you please explain the notations $\hat X$ and $\hat Y$? | |
| Nov 6, 2022 at 17:57 | history | asked | Mira | CC BY-SA 4.0 |