Talk:Frobenius algebra

Expanded

I expanded the article a bit and gave a good reference for history and algebraic study of Frobenius algebras. In particular, everything up to the categorical stuff is easily found in Lam (which has several sections on Frobenius algebras), and everything after is unsourced, but I remember some Baez article about this.

I did not actually include the duality theory which plays a prominent role in the study of Quasi-Frobenius rings and Iwanaga-Gorenstein rings. In particular, I did not include the classical contributions of Nakayama (because he has no wiki bio!) on the Nakayama permutation, the relations amongst the simple modules, and socle/head duality. Nor did I include Nakayama's term for these algebras, Frobeniusean, since he is the only one who used it more than once. Similarly, symmetric algebras received almost no special treatment, even though they are a fundamentally important special case. The generalizations to Frobenius rings, Quasi-Frobenius rings, and self-injective rings are not included here, but perhaps this would be a good place if quasi-frobenius does not get its own article. Lam would provide a good reference for the intuitive differences between these notions, but for generalizations of QF, one needs to look elsewhere (Faith's Rings and Things for instance has quite a few such generalizations). JackSchmidt (talk) 09:19, 23 November 2007 (UTC)

Lam also acts as a source for the TQFT remarks, but has nothing on the categorical definition. I included a few other remarks too. JackSchmidt (talk) 09:55, 23 November 2007 (UTC)

Characeristic of the field?

Is it correct that there's no requirement on the characteristic of the field k in the definition of Frobenius algebras? HSNie 14:56, 31 May 2013 (UTC) — Preceding unsigned comment added by HSNie (talk • contribs)

TODO list

Some topics need expansion:

• Need defintion of a dagger frobenius algebra. This algebra is used to define the basis in a dagger compact category.
• The dagger frobenius algebras somehow convey classical information, because they can evade the no-colning/no-deleting theorems. This should be clarified.
• Need to cleanup the statements about TQFT, so, e.g. Atiyah says any TQFT is just a functor between nCob and FDHilb both of which are dagger compact. This article never goes there. so ???

User:Linas (talk) 05:09, 28 November 2013 (UTC)

• Need to explain how the categorical definition is a generalization of the original one. In exactly what category is a frobenius object a frobenius algebra? Possibly give examples of interesting frobenius objects in other categories. — Preceding unsigned comment added by Wikidsp (talk • contribs) 00:04, 10 August 2016 (UTC)

why finite-dimensional?

Naively, trace-class operator algebras would be seem to fit the profile; why the restriction to fininte? User:Linas (talk) 06:28, 28 November 2013 (UTC)

136.159.16.10 (talk) 20:12, 16 July 2018 (UTC) They are necessarily finite because the unitality of the Frobenius law makes the snake equations (for monoidal categories) hold, which is only possible in finite dimensions. Compose the unit with the comultiplication and the counit with with the multiplication and see for yourself.

Place within mathematics curriculum

Could someone please expand upon where Frobenius algebra would be covered in a typical mathematics curriculum? — Preceding unsigned comment added by 67.52.192.26 (talk) 15:13, 20 June 2014 (UTC)