I have adjusted the wording in the Idea-section (here) and then considerably expanded (compared to what was there before), adding paragraphs on the two roles that Frobenius algebras play in quantum physics (2d TQFTs and quantum measurement)

]]>I have added text (here) to show the slightly different conventions for “special Frobenius” used in the literature.

For what it’s worth, a Google search for

```
Frobenius algebra "called special"
```

gives me the top hit, the second & the third all agreeing that the condition is “$prod \circ coprod = id$”, while the FRS definition shows up only in later hits.

In any case, the entry should list all conventions instead of silently picking one — which it does now, but please be invited to add further references etc.

]]>(followup to #33) What reference are we using in Frobenius algebra for special FA’s? In Definition 2.22.iii of FFRS06 they define a special FA as a FA s.t. $\mu\Delta=\beta_A\text{id}_A$ and $\eta\epsilon=\beta_1\text{id}_1$. In p.5 of Yadav22 they use the same definition for special FA but also state that a FA only satisfying the first condition is a separable Frobenius algebra.

]]>added a Properties-subsection (here) on “Normal form and ’Spider theorem”“

]]>added (here) the example of the Frobenius algebras induced by a linear basis on a vector space, together with some commentary on its role in quantum information theory

]]>Clarified the definition of Frobenius algebras in the category of Vectᴋ by making the non-degenerate pairing more explicit. The previous expression was the map u ↦ ϵ∘μ(1 ⊗ u); however, interpreting the symbols on the right-hand side completely literally in their syntactic form, then by the axioms of a unital associative we have that ϵ(μ(1 ⊗ u)) = ϵ(u) ∈ K. We are trying to have the right-hand side land in V*, which is done by clarifying that the right-hand side should actually be v ↦ ϵ∘μ(v ⊗ u), which is indeed an element of V* and the desired isomorphism of V with V*.

Lillian Ryan Uhl

]]>Added a note that the usual two Frobenius laws follow from the single axiom $(1 \otimes \mu) \circ (\delta \otimes 1) = (\mu \otimes 1) \circ (1 \otimes \delta)$.

]]>Thanks! At least they give it a notation, $\beta_1(A)$.

I notice that their definition of “special Frobenius algebra” is different than the one at Frobenius algebra: we require $\mu\circ\delta = 1$, whereas they require that $\mu\circ\delta$ and $\epsilon\circ\eta$ are a nonzero multiple (in their $\mathbb{C}$-linear context) of the identity. I guess the relationship is that in the latter case, one can multiply $\epsilon$ by the invertible scalar $\mu\circ\delta$ and divide $\delta$ by the same scalar to get another Frobenius structure on the same underlying algebra in which $\mu\circ\delta = 1$ as in our definition, but that still leaves the condition that $\epsilon\circ\eta$ is invertible – is that somehow implied by our definition?

]]>Some such composition occurs, but unnamed, e.g. Fig. 11 of this.

]]>Any Frobenius algebra $A$ has an invariant induced by composing the unit of the multiplication with the counit of the comultiplication, $I \xrightarrow{\eta} A \xrightarrow{\epsilon} I$. What is this called? Is it ever interesting?

]]>Isn’t it? I’m not sure quite what to make of it.

]]>Oh, hey, that’s pretty neat!

]]>I added a bit more, including a claim that the free polycategory containing a Frobenius algebra is the terminal polycategory.

]]>Yes, I went ahead and did it, but then got called away before I could announce it. I also incorporated a couple of “TODO”s from the bottom of the page into Frobenius algebra, and started adding a bit about Frobenius monoids in polycategories.

]]>It looks like this has already been done, but sounds good to me.

]]>I think the first version is probably best.

]]>We have a stubby page Frobenius monoid that seems to be intended to be about an abstract categorical version of a Frobenius algebra. However, much of the page Frobenius algebra is already written in the generality of an arbitrary monoidal category. So I think we should either get rid of Frobenius monoid and redirect it to Frobenius algebra (and add a bit of discussion), or else try to separate the abstract from the concrete versions. My inclination is the former; any other opinions?

]]>Thanks! I have included that at *Cardy condition*.

Ok, I followed the bizarre process for uploading an image (in the Sandbox).

cardy_condition.jpg, screen captured from Lauda & Pfeiffer 2006

I did this on my machine with Adobe Acrobat at 150% magnification which might give different sized bitmaps on machines with different screens. If you think this image should be a different size, or want more of the related images, Capturing a screen dump is trivial, but you also need something simple like MS-Paint to crop and save the image.

]]>Please do! Upload the image to the nLab itself, as described here. Thanks.

]]>So, Urs, see if Aaron has the xypic code still around.

The xypic code is in the TeX source in the ArXiv.

However John probably just “screen captured” (print screen) from a display of the PDF and used something like MS-Paint to crop down the bitmap and save it as a JPEG. I’d do it for you but I don’t have convenient place to host the image for upload into the nLab.

]]>The Cardy condition seems to be in here

Look at 1.14.

So, Urs, see if Aaron has the xypic code still around.

]]>Back then some kind soul provided these cobordism pictures at Frobenius algebra. Is that somebody still around and might easily provide also the picture for the Cardy condition?

I assume you mean such things pictures as commutative_law.jpg (IS THERE ANY WAY TO EMBED IMAGES IN THE NFORUM?)

Those pics come from

How is what you want different?

]]>Back then some kind soul provided these cobordism pictures at *Frobenius algebra*. Is that somebody still around and might easily provide also the picture for the Cardy condition?