Timeline for Para-Complexification of Lie Groups
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 21, 2014 at 17:11 | vote | accept | CommunityBot | ||
| Oct 21, 2013 at 13:15 | history | edited | user21574 |
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| Oct 19, 2013 at 2:03 | comment | added | JHM | The above definition of paracomplexity is bizarre. The linear isomorphism $K:V\to V$ already splits over $\mathbb{R}$, i.e. $V=V_+ \oplus V_-$. So what is the meaning of saying ``the paracomplexification of $(V,K)$ is $V\otimes \mathbb{C}$"? | |
| Oct 19, 2013 at 1:51 | comment | added | JHM | The above definition of $complexification$ is not the standard notion. Typically, for a real lie group $G$ one complexifies with respect to a maximal compact subgroup $K$ of $G$. The lie algebra of the complexification $G_{K,\mathbb{C}}=G_\mathbb{C}$ will be identified with $\mathfrak{k} \otimes_\mathbb{R} \mathbb{C}$. Without assuming that $G$ is reductive I am skeptical that a `complexification" in the sense of the above mapping property exists or is unique up to biholomorphism. | |
| Oct 18, 2013 at 17:44 | answer | added | Ben McKay | timeline score: 1 | |
| Oct 18, 2013 at 14:59 | history | edited | user21574 |
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| Oct 16, 2013 at 17:37 | history | edited | user21574 |
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| Oct 15, 2013 at 20:11 | comment | added | Paul Reynolds | An almost-paracomplex manifold is a smooth real even-dimensional manifold with a paracomplex structure (an endomorphism defined as above) defined on its tangent bundle, and morphisms are smooth maps preserving it. I can't remember if there is a notion of integrability. | |
| Oct 15, 2013 at 15:21 | comment | added | S. Carnahan♦ | There are several undefined steps here, since you haven't defined what a paracomplex manifold is, or what the analogue of holomorphic map should be. However, in the linear category, it is clear that the paracomplexification of the Lie algebra is given by taking the direct sum of two copies. When your Lie group is compact (e.g., for your example of $U(n)$), you can use Chevalley's theorem to transport the question into the setting of anisotropic reductive groups: see mathoverflow.net/questions/6079/… | |
| Oct 15, 2013 at 14:07 | comment | added | Ben McKay | I think the paracomplexification is $G \times G$. | |
| Oct 15, 2013 at 9:29 | history | edited | user21574 |
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| Oct 15, 2013 at 9:02 | comment | added | user21574 | You can find here digital.csic.es/bitstream/10261/15773/1/RockyCFG.PDF . | |
| Oct 15, 2013 at 4:46 | comment | added | Peter Crooks | What would it mean for a Lie group $G$ to admit a para-complex structure? Is there a notion of integrability analogous to that required for an almost complex structure to be complex? | |
| Oct 14, 2013 at 21:13 | history | asked | user21574 | CC BY-SA 3.0 |