Timeline for Invariant Vector Fields for Homogenous Spaces
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2018 at 9:01 | comment | added | R. Rankin | I like your answer I just wanted to address the (most? all?) (often? always?) remarks that your put in there (: | |
| Nov 25, 2018 at 2:44 | comment | added | Jason DeVito - on hiatus | @R.Rankin. Yes, for Lie groups, the tangent space at each point is spanned by left invariant vector fields. My answer shows that this does not occur in general for homogeeous spaces (and note that a Lie group is very special kind of homogeneous space.) | |
| Nov 24, 2018 at 1:52 | comment | added | R. Rankin | @JasonDeVito I'm pretty sure that the Left-invariant vector fields on $S^3$ (aka $SU(2)$ for our purposes) span the space. However, I'm thinking that the physics definition of homogenous space might be different from the math one? | |
| Mar 22, 2010 at 20:44 | comment | added | Aston Smythe | Well, as a grad student I really like "Grad student/homework" answers. | |
| Mar 22, 2010 at 13:22 | comment | added | Jason DeVito - on hiatus | @Ben: Yes, that's a great summary! I guess that as a grad student, I still write in the "Grad student/homework" mode with significantly more detail than is usually presented in papers (or on MO). See, for example, most of my other posts ;-). | |
| Mar 22, 2010 at 13:20 | comment | added | Jason DeVito - on hiatus | @Aston: No, the point is that for G-invariance, the only invariant vector field is typically the 0 field. For H-invariance, it's usually (always?) the case, that they don't span at even a single point. | |
| Mar 22, 2010 at 12:27 | comment | added | Aston Smythe | Can one construct a basis of all vector fields using the invariant ones, like in the Lie group case? | |
| Mar 22, 2010 at 1:20 | comment | added | Ben Wieland | In other words: the tangent vector space to $G/H$ at $eH$ is the quotient of the Lie algebra of $G$ by the Lie algebra of $H$. $G$ acts on its Lie algebra by the adjoint action and the restriction of this to $H$ descends to act on the quotient vector space. The vectors fixed under this action are exactly the ones that extend to invariant vector fields. | |
| Mar 21, 2010 at 20:22 | history | answered | Jason DeVito - on hiatus | CC BY-SA 2.5 |