Timeline for Primitive elements in group hopf algebras over fields of non-zero characteristic
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 15, 2016 at 10:14 | answer | added | Ehud Meir | timeline score: 6 | |
| Sep 14, 2016 at 21:15 | comment | added | Konstantinos Kanakoglou | I am not sure that your argument is valid for non-zero characteristic. However, maybe I am wrong. I need to think a little more about it and maybe I will come back. In any case thank you for your prompt response! | |
| Sep 14, 2016 at 21:08 | comment | added | darij grinberg | Yes. If at least one $h \neq 1$ satisfies $c_h \neq 0$, then $\Delta\left(\sum_{g \in G} c_g g\right)$ has a nonzero coefficient in front of $h \otimes h$, whereas $\left(\sum_{g \in G} c_g g\right) \otimes 1 + 1 \otimes \left(\sum_{g \in G} c_g g\right)$ does not. So the only elements that have a chance to be primitive are those of the form $c_1 1$. But those can only be primitive if $c_1 = 0$. | |
| Sep 14, 2016 at 21:07 | comment | added | Konstantinos Kanakoglou | @darij grinberg: even over a field of characteristic $2$? | |
| Sep 14, 2016 at 20:59 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
edited title
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| Sep 14, 2016 at 20:54 | comment | added | darij grinberg | The primitive elements of a group Hopf algebra are always $0$. This is easy to check by computing the coproduct of $\sum_{g \in G} c_g g$ and comparing it with $\left(\sum_{g \in G} c_g g\right) \otimes 1 + 1 \otimes \left(\sum_{g \in G} c_g g\right)$. | |
| Sep 14, 2016 at 20:49 | comment | added | Konstantinos Kanakoglou | Thanks for mentioning this. It would be interesting if you find some time to add some details and turn your comment into an answer. However, I am mainly interested into some group hopf algebra example. | |
| Sep 14, 2016 at 20:37 | comment | added | Qiaochu Yuan | An easy example is $k[x]/x^p$ where $k$ has characteristic $p$ and $x$ is primitive. This is the group scheme $\alpha_p$ whose functor of points sends a commutative $k$-algebra $A$ to the additive group of $a \in A$ such that $a^p = 0$. | |
| Sep 14, 2016 at 20:27 | comment | added | Konstantinos Kanakoglou | this is similar to question math.stackexchange.com/questions/1924526/…, which however received no feedback. | |
| Sep 14, 2016 at 20:26 | history | asked | Konstantinos Kanakoglou | CC BY-SA 3.0 |