Timeline for Cohomology of linear algebraic groups
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9 events
| when toggle format | what | by | license | comment | |
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| Sep 18, 2019 at 11:48 | vote | accept | CommunityBot | ||
| Sep 17, 2019 at 17:43 | answer | added | skupers | timeline score: 4 | |
| Sep 17, 2019 at 17:29 | comment | added | YCor | For the continuous one, it's a standard fact about central extensions of semisimple Lie groups. For the non-continuous one, it's related to (symplectic) K-theory and more complicated (oh, well I'm sure that $H_2$ is huge; that $H^2$ valued in $\mathbf{C}$ is still huge, should be checked); see mathoverflow.net/questions/63529 for a related discussion. | |
| Sep 17, 2019 at 16:32 | comment | added | user39380 | @YCor Thanks for pointing out, let's focus on continuous cohomology. And would you explain the calculation of the two versions of $H^2(\mathrm{Sp}_{2n}(\mathbb{C}),\mathbb{C})$? Thanks! | |
| Sep 17, 2019 at 16:00 | comment | added | YCor | As regards (2), do you mean group cohomology or continuous group cohomology? For instance $H^2(\mathrm{Sp}_{2n}(\mathbf{C}),\mathbf{C})$ is huge but the continuous cohomology is trivial. Maybe the same phenomenon occurs with the module $\mathbf{C}^{2n}$, I'm not sure. In general guess that the continuous cohomology in degree $\ge 1$ of any finite-dimensional $G$-module is zero, for $G$ semisimple. | |
| Sep 17, 2019 at 15:54 | comment | added | YCor | @Bombyxmori If it's isomorphic to a Lie algebra cohomology group, this is a theorem, not a definition. In the $\mathbf{Z}$ case this sounds quite doubtful, by the way. | |
| Sep 17, 2019 at 15:53 | comment | added | Bombyx mori | Isn't that defined by the cohomology of the Lie algebras? I think Chevally studied it during 1950s... | |
| Sep 17, 2019 at 15:53 | history | edited | YCor |
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| Sep 17, 2019 at 15:49 | history | asked | user39380 | CC BY-SA 4.0 |