Timeline for How to interpret compositional diagrams for quantum sets algebraically
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 15, 2021 at 20:33 | comment | added | Ben A-S | Okay, thank you. | |
| Dec 15, 2021 at 8:36 | comment | added | Matthew Daws | I also find it hard to work "just with diagrams". Vicary showed in arxiv.org/abs/0805.0432 that SSFA's are just finite-dimensional $C^*$-algebras equipped with a trace satisfying a certain normalisation condition (which I always forget...) The trace turns the $C^*$-algebra into a Hilbert space, and $\delta$ is the the Hilbert space adjoint of the multiplication map. This perspective makes it clear that $\delta(1)=1\otimes 1$ only for trivial $A$. | |
| Dec 15, 2021 at 6:37 | history | edited | YCor | CC BY-SA 4.0 |
formatting
|
| Dec 15, 2021 at 6:30 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
added paper details and link
|
| Dec 15, 2021 at 2:58 | comment | added | Ben A-S | Is it true that a Frobenius algebra is trivial if and only if $\delta(1) = 1\otimes 1$? | |
| Dec 15, 2021 at 2:34 | comment | added | Ben A-S | Oh, wow, yes, you're absolutely right. I'm mixing up the co-algebra and algebra structures. Thanks for your help! | |
| Dec 14, 2021 at 23:28 | comment | added | Jules Lamers | .. And I suppose that in that way Thm 2.6 gives an inner product, allowing us to switch between elements and their duals. I think that on the right in (16) the author omitted the little 'popsicle diagram' for the unit dangling at the bottom in (15). So for Q2 I think that the relation says: $f^\dag(x) = (f(x^\dag))^\dag$. | |
| Dec 14, 2021 at 23:13 | comment | added | Jules Lamers | For Q1: a Frobenius algebra is not a bialgebra, so I'm not sure if $\delta(1)$ equals $1\otimes1$ (which the mixes algebra and coalgebra structures). In Sweedler notation $\delta(1)=\sum 1_{(1)} \otimes 1_{(2)}$ so we seem to get $a\mapsto \sum a^\dag(1_{(2)}) \, 1_{(1)}$ which still $\in A$. [Also, I would think the last diagrams in (12) and (17) ought to be reflected] | |
| Dec 14, 2021 at 22:52 | history | asked | Ben A-S | CC BY-SA 4.0 |