Timeline for Comultiplication of elements of partition of unity
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Feb 25, 2020 at 23:44 | history | suggested | JP McCarthy | CC BY-SA 4.0 |
Just noting that the question has changed since this was asked.
|
| Feb 25, 2020 at 18:38 | review | Suggested edits | |||
| S Feb 25, 2020 at 23:44 | |||||
| Feb 25, 2020 at 18:25 | comment | added | JP McCarthy | mathoverflow.net/a/353545/35482 | |
| Feb 4, 2020 at 13:59 | comment | added | JP McCarthy | I am trying to construct an explicit counterexample for commutative $F(G)$, i.e. a finite group $G$. This math.stackexchange.com/questions/3534041/… points the way to a counterexample. Otherwise I can construct a homomorphism from a finite group to $\mathbb{Z}_d$ via $\text{supp }p_i\rightarrow\{i\}$ and the kernal of this is $S_0:=\text{supp }p_0\rhd G$, and $G/S_0\cong C_d$ and the formula for the $\Delta(p_i)$ follows, and there is then no counterexample for commutative $G$. There surely is a counterexample. | |
| Jan 30, 2020 at 0:03 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
added 4 characters in body
|
| Jan 27, 2020 at 19:32 | comment | added | JP McCarthy | I will get back in detail on Wednesday | |
| Jan 27, 2020 at 19:21 | comment | added | Konstantinos Kanakoglou | @JP McCarthy, i am not sure i have understood your last comment. Notice however , that if you take as $F(G)$ the dual of a group hopf algebra (as in my examples above) then it will always be commutative (since the group hopf algebra is always cocommutative). | |
| Jan 27, 2020 at 18:48 | comment | added | JP McCarthy | I don't think a $F(G)$-commutative counterexample is possible. I reckon in this case $p_0$ is the indicator function on a normal subgroup $H$, and the $p_i$ the indicator function on the cosets and $G/H\cong C_d$ and the formulae for $\Delta(p_i)$ follows. I think I can show this with the homomorphism $\operatorname{supp} p_i\rightarrow i\in C_d$; will get back to it Wednesday. I am possibly missing the condition that for all projections $q\in F(G)$, there exists $k_q$ such that $\nu^{\star k_q}(q)\neq 0$. That is an assumption I am missing (but can possibly do without). | |
| Jan 23, 2020 at 18:45 | vote | accept | JP McCarthy | ||
| Feb 25, 2020 at 18:23 | |||||
| Jan 23, 2020 at 17:39 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
added 30 characters in body
|
| Jan 23, 2020 at 17:35 | comment | added | Konstantinos Kanakoglou | On the other hand, if we consider both $N$, $H$ to be non-abelian then none of their dual hopf algebras $(kN)^*$, $(kH)^*$ (which are both hopf subalgebras of $F(G)$) will be cocommutative; this is the situation in the counterexample $F(G)=\big(k(N\times H)\big)^*\cong(kH)^*\otimes(kN)^*$. | |
| Jan 23, 2020 at 17:31 | comment | added | Konstantinos Kanakoglou | and since it is also commutative, then it will be isomorphic to a (self-dual) group hopf algebra for a finite abelian group. so it will be $kC_d$ (or $kH$ fore some finite abelian group). this corresponds to my first example (notice that $F(G)=\big(k(N\times C_d)\big)^*\cong kC_d\otimes (kN)^*$ is generally not cocommutative if $N$ is not abelian). | |
| Jan 23, 2020 at 16:34 | comment | added | Konstantinos Kanakoglou | @JP McCarthy, i am not claiming that the comultiplication is cocommutative on $F(G)$. (I guess my introductory comment was misleading). I was just refering to the set of the $d$ orthogonal idempotents: what i had in mind, was that if your comultiplication is valid then the $d$ idempotents span a hopf subalgebra. This is the cocommutative one. | |
| Jan 23, 2020 at 9:13 | history | bounty ended | JP McCarthy | ||
| Jan 23, 2020 at 9:13 | comment | added | JP McCarthy | I don't know why you think the comultiplication is cocommutative? If it is cocommutative on these $d$ elements I am not sure why it follows more generally (but this is by-the-by). I must construct a small example and see how the counterexample works. Thank you. | |
| Jan 23, 2020 at 3:31 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
added 14 characters in body
|
| Jan 23, 2020 at 3:21 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
added 88 characters in body
|
| Jan 23, 2020 at 3:12 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
added 14 characters in body
|
| Jan 23, 2020 at 2:33 | history | answered | Konstantinos Kanakoglou | CC BY-SA 4.0 |