Question

One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V → k as the co-unit (or equivalently, tr(a⋅b) as the frobenius form). This is enough data to generate a comultiplication δ : V → V ⊗ V. This turns out to be μ^†, for multiplication μ. Is there any intuition for what this map does (aside from the obvious "do multiplication on the dual space")?

Answer 1

Here's how I live to think about matrices. Penrose (1971) figured out that you can draw linear algebra diagrammatically. A picture in the Penrose notation is a directed labeled graph with external leaves. The edges are labeled by vector spaces (changing the direction on an edge has the same effect as swapping the label X with the dual vector space X*), and vertices by multilinear maps. In this way, placing two edges next to each other is the tensor product. The ground field R should be drawn as an invisible edge, so that X ⊗ R = X.

So, pick your favorite finite-dimensional vector space X, and think about the types of diagrams you can draw using just it. Well, the space of matrices (what you call V) is X ⊗ X*, so it looks like two parallel lines pointed in opposite directions. Then you can check that the trace is the directed cap, the identity element (thought of as a map R → V) is the directed cup, and multiplication and comultiplication are both given by trivalent vertices.

In ASCII (ignore the weird coloring):

       |            |                        | |
       |            |                        | |
 X  =  ^ ,   X*  =  v ,  R = [empty],  V  =  ^ v
       |            |                        | |
       |            |                        | |


          ->-            |     |
         /   \           ^     v
 Tr  =  |     |    I  =  |     |
        ^     v           \   /
        |     |            -<-


            | |                  | |   | |
            ^ v                  ^ v   ^ v
            | |                  | |   | |
 mu  =     / _ \       delta  =   \ \_/ /
          / / \ \                  \   /
         | |   | |                  | |
         ^ v   ^ v                  ^ v
         | |   | |                  | |

Not only does the notation "explain" the comultiplication, it "proves" all the associativity and unital properties you might want. Mostly, though, I think it makes it totally clear what the Frobenius pairing (a,b) → Tr(ab) is doing. It's just the map:

              ->-
             / _ \ 
 pair =     / / \ \ 
           | |   | | 
           ^ v   ^ v 
           | |   | |

Which is just the canonical fact that (X ⊗ X*)* = X ⊗ X*. This ability to rotate X ⊗ X* is why δ = μ*.