Timeline for Questions Suggested by the Parabolic Subgroup Definition
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 9, 2010 at 3:38 | comment | added | Jean Delinez | Let me try again: For a general semi-simple compact Lie group $G$ with Lie algebra $\mathfrak{g}$, and a parabolic subgroup $P$ with corresponding Lie subalgebra $\mathfrak{p}$, is there a nice description of the algebra of continuous functions of $G$ in terms of $\mathfrak{g}$ or $\mathbb{p}$, or both? | |
| Feb 8, 2010 at 22:15 | comment | added | Mike Skirvin | I'm not sure if I understand. $G/P$ isn't a group, so I don't know what is meant by its Lie algebra. Perhaps I misunderstood the question? | |
| Feb 8, 2010 at 21:47 | comment | added | Mike Skirvin | All quotients of parabolics are of the form you list in the case of $G = U(n),$ but in general there is no "flag-like" description of partial flag-varieties for general $G.$ For the classical groups, however, there are flag-like descriptions, and it's fun finding the Borels and parabolics that they correspond to. As to your last question, there should be lots of examples. You mention elliptic curves, and higher genus curves should also fall into this category. Also, as far as I know, the Schubert subvarieties of flag varieties aren't a quotient of linear algebraic groups. | |
| Feb 8, 2010 at 18:21 | comment | added | Jean Delinez | (iii) So all quotients of complex linear algebraic groups by subgroups containing the Borel subgroup are of the form $U(n)/U(k_1) \times \cdots \times U(k_m)$? When the quotient is by the Borel subgroup, is this reflected in some simplification of $U(k_1) \times \cdots \times U(k_m)$? (iv) Hopefully this is not too bad practice, but I'm going to generalise my last question and ask: What are examples (besides elliptic curves) of complex projective varieties that aren't quotients of comapct semi-simple complex Lie groups by compact subgroups? | |
| Feb 8, 2010 at 16:28 | history | answered | Mike Skirvin | CC BY-SA 2.5 |