Timeline for Generalization of the flatness of $\mathbb R^3$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2023 at 15:09 | comment | added | Ben McKay | For a counterexample, you can take $G$ a torus, $H$ a subtorus, and $\mathfrak{n}\subset\mathfrak{g}$ a linear subspace, complementary to the Lie algebra $\mathfrak{h}$ of $H$, but so that $\mathfrak{n}$ is the tangent space at $1\in G$ of a dense subgroup of $G$. | |
| Aug 29, 2023 at 10:53 | comment | added | A. J. Pan-Collantes | By the way, I guess that there are examples when this semidirect sum is not coming from the existence of a semidirect product structure in $G$... | |
| Aug 29, 2023 at 10:49 | comment | added | A. J. Pan-Collantes | @BenMcKay I see, my requirement can be weakened to: $(G,H)$ is such that $\mathfrak{g}$ is decomposed as the semidirect sum of a "given" Lie subalgebra $\mathfrak{n}$ and the Lie subalgebra $\mathfrak{h}$. Anyway, I prefer my formulation, because (for me) the idea of semidirect product is nearer to intuition. | |
| Aug 29, 2023 at 9:03 | comment | added | Ben McKay | $G$ doesn't have to be a semidirect product. The Lie algebra is a semidirect sum. | |
| Aug 25, 2023 at 6:35 | history | answered | A. J. Pan-Collantes | CC BY-SA 4.0 |