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There is an equivalence between the category of commutative finite dimensional Frobenius algebras and 2 dimensional topological quantum field theories, see for example the book by Joachim Kock, which I read a bit (but I have no idea about the topological part).

Some questions:

  1. Do module theoretic notions have a topological interpretation using this equivalence? For example an open problem for commutative finite dimensional Frobenius algebras A is wheter $Ext_A^1(M,M) \neq 0$ for any non-projective module $M$. Does this have a topological interpretation?

  2. Given a commutative f.d. Frobenius algebra $A$, then its trivial extension algebra $T(A)$ (see for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra for the definition) is again a commutative Frobenius algebra with twice the dimension of $A$. What is the topological interpretation of the trivial extension construction using the equivalence?

  3. For two f.d. commutative Frobenius algebras $A,B$ one has $rad(A \otimes B)=rad(A) \otimes B + A \otimes rad(B)$ (here rad dentotes taking the Jacobson radical of an algebra). This reminds me of the formula for the boundary of the product of two topological spaces. Is this just random, or is there a connection between those two formulas?

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Maybe you are not so interested in this, but there is a nice physical interpretation of modules over a Frobenius algebra.

(1+1)d TQFTs are used to describe topological phases of matter in 1d. Think of a sequence of spin 1/2 particles connected up in a line or a circle where the global state cannot be specified by local observables. Unfortunately, without more structure, topological phases of matter in 1d are trivial. This is a manifestation of the fact that any frobenius algebra is isomorphic to a product of matrix algebras. In this setting, bi-modules correspond to domain walls between different topological phases of matter. Since topological phases of matter are boring in 1d, this isn't really saying much.

In 2d things are much more interesting. Topological phases of matter are now classified by (2+1)d TQFTs which can be specified by fusion categories and slight generalizations of them. There are a lot of interesting fusion categories and domain walls between the corresponding topological phases of matter are specified by bi-modules over fusion categories.

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    $\begingroup$ I do not understand much of this but why are Frobenius algebras a product of matrix algebras (you mean matrix algebras over a field)? This is not true. For example $K[x]/(x^2)$ is not a product of matrix algebras over a field. $\endgroup$
    – Mare
    Sep 19, 2017 at 6:34
  • $\begingroup$ It is possible that I am not using the correct adjectives, but I mean something like this math.ucr.edu/home/baez/qg-winter2001/qg16.1.html $\endgroup$
    – Daniel Barter
    Sep 19, 2017 at 23:27
  • $\begingroup$ in the linked context--the artin-wedderburn therem--it would be matrix algebras over a division ring (non-commutative field) $\endgroup$
    – Samantha Y
    Sep 28, 2017 at 18:15

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