What is the comultiplication of a matrix frobenius algebra? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:08:29Z http://mathoverflow.net/feeds/question/2802 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2802/what-is-the-comultiplication-of-a-matrix-frobenius-algebra What is the comultiplication of a matrix frobenius algebra? Aleks Kissinger 2009-10-27T11:41:43Z 2009-10-31T23:38:07Z <p>One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V &rarr; k as the co-unit (or equivalently, tr(a&sdot;b) as the frobenius form). This is enough data to generate a comultiplication &delta; : V &rarr; V &otimes; V. This turns out to be &mu;<sup>&dagger;</sup>, for multiplication &mu;. Is there any intuition for what this map does (aside from the obvious "do multiplication on the dual space")?</p> http://mathoverflow.net/questions/2802/what-is-the-comultiplication-of-a-matrix-frobenius-algebra/2894#2894 Answer by Theo Johnson-Freyd for What is the comultiplication of a matrix frobenius algebra? Theo Johnson-Freyd 2009-10-27T20:59:56Z 2009-10-31T23:38:07Z <p>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 <i>X</i> with the dual vector space <i>X</i>*), and vertices by multilinear maps. In this way, placing two edges next to each other is the tensor product. The ground field <strong>R</strong> should be drawn as an invisible edge, so that <i>X</i> &otimes; <strong>R</strong> = <i>X</i>.</p> <p>So, pick your favorite finite-dimensional vector space <i>X</i>, and think about the types of diagrams you can draw using just it. Well, the space of matrices (what you call <em>V</em>) is <i>X</i> &otimes; <i>X</i>*, 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 <strong>R</strong> &rarr; <em>V</em>) is the directed cup, and multiplication and comultiplication are both given by trivalent vertices.</p> <p>In ASCII (ignore the weird coloring):</p> <pre><code> | | | | | | | | X = ^ , X* = v , R = [empty], V = ^ v | | | | | | | | -&gt;- | | / \ ^ v Tr = | | I = | | ^ v \ / | | -&lt;- | | | | | | ^ v ^ v ^ v | | | | | | mu = / _ \ delta = \ \_/ / / / \ \ \ / | | | | | | ^ v ^ v ^ v | | | | | | </code></pre> <p>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) &rarr; Tr(ab) is doing. It's just the map:</p> <pre><code> -&gt;- / _ \ pair = / / \ \ | | | | ^ v ^ v | | | | </code></pre> <p>Which is just the canonical fact that (<i>X</i> &otimes; <i>X</i>*)* = <i>X</i> &otimes; <i>X</i>*. This ability to rotate <i>X</i> &otimes; <i>X</i>* is why &delta; = &mu;*.</p>