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These days, lots of people are excited by Frobenius algebras because commutative Frobenius algebras are the same thing as 2D topological quantum field theories.

...but this seems like teaching an old dog new tricks. Can anyone sum up (using only diet representation theory :-P), why Frobenius algebras were invented and what was so good about them?

Also, any nice texts and/or papers along this line would be much appreciated. I'm working through the old Nakayama papers now, but perhaps this material exists in a friendlier and more modernised form somewhere?

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Because group algebras are Frobenius algebras? –  Bruce Westbury Jul 16 '10 at 16:52
The notion of "Frobenius algebra" has evolved a lot, toward for example Frobenius objects in monoidal categories. The early notion arose when people started to explore group algebras of finite groups and more generally finite dimensional or artinian algebras with similar properties. Not a central issue at first. Then a basic result of Larson-Sweedler (every finite dimensional Hopf algebra is Frobenius) led further into Hopf algebras. Related duality ideas arose in geometric/topological settings. No single source is adequate now, but the books by Curtis-Reiner go beyond Nakayama. –  Jim Humphreys Jul 16 '10 at 17:04
Curtis and Reiner is a comprehensive reference, in spite of its age. For a more recent if less detailed introduction, see Drozd and Kirichenko, Finite-dimensional algebras. –  Victor Protsak Jul 17 '10 at 2:09
C & R looks like exactly the thing I've been looking for! Thanks for the input guys. –  Aleks Kissinger Jul 17 '10 at 16:58
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2 Answers

up vote 5 down vote accepted

Frobenius's original turn-of-the-century perspective was the nonvanishing of a determinant. Brauer–Nesbitt–Nakayama studied some equivalent definitions in the late 30s and early 40s. For instance, an equivalence between the left and right regular representations is a rare and beautiful thing; this gives an equivalence between projectivity and injectivity that is explained in modern language in Lam's Lectures on Modules and Rings. This also gives a "perfect duality" studied by Dieudonné in the late 50s. I added the missing early sources to the wikipedia article, including the Brauer–Nesbitt announcement in PNAS which is pretty easy to read.

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Brauer's student Nesbitt also defined a special class of finite dimensional Frobenius algebras over fields called (unfortunately) symmetric: for this you need a nondegenerate bilinear form which is both associative and commutative. Group algebras of finite groups all have this property (visible in the symmetry of the matrix $C$ of Cartan invariants in prime characteristic); but finite dimensional Hopf algebras such as restricted enveloping algebras of $p$-Lie algebras need not be symmetric algebras. The early work was done in a purely algebraic study of foundations, I think. –  Jim Humphreys Jul 16 '10 at 20:14
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I learned the following from Constantin Teleman, and from these lectures of David Ben-Zvi.

Let $G$ be a finite group. Let $A = \mathbb{C}[G]^G$. This is an algebra under convolution:

$$(\phi \cdot \psi)(g) := \sum_{h \in G}\phi(gh^{-1})\psi(h).$$

We also have a trace $t : A \to \mathbb{C}$ given by $t(\phi) = \phi(1)/|G|$.

This is a Frobenius algebra. (Maybe this is the original example of a Frobenius algebra?)

In fact, this is a semisimple Frobenius algebra: Let $P_i = \frac{\chi_i \operatorname{dim}\chi_i}{|G|}$, where $\chi_i$ are the irreducible characters. Then $P_i$ is a basis for $A$ and $P_i\cdot P_j = \delta_{ij}P_i$.

Note that $1 = \sum_i P_i$. We have $t(1) = t(\sum_i P_i) = \frac{\sum_i (\operatorname{dim} \chi_i)^2}{|G|^2}$. On the other hand we have $t(1) = 1/|G|$. We get the formula $$|G| = \sum_i (\operatorname{dim} \chi_i)^2.$$

Pretty cool, no?

I think the correspondence between Frobenius algebras and 2D TQFTs is more than just "teaching an old dog new tricks". The 2D TQFT corresponding to this Frobenius algebra is the "finite group ('pure') gauge theory" of Dijkgraaf-Witten. From the TQFT perspective, $t(1)$ is the number which is assigned to the sphere $S^2$. It's some kind of "path integral"(?) over all maps $S^2 \to BG = \ast/G$. This number is a certain "weighted count" of $G$-bundles over $S^2$. And of course we have a similar story for all other surfaces. When the surfaces have boundary, we look at $G$-bundles with prescribed holonomy around the boundary circles.

Moreover, the 2D TQFT can be "extended" to manifolds with corners, which is also interesting. Check out Ben-Zvi's lecture notes for more on this stuff.

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I think the point of Aleks' question is not that 2D TQFTs aren't interesting, but that they couldn't possibly have been the original motivation for the definition. –  Qiaochu Yuan Jul 16 '10 at 21:31
Okay, my bad. I just get too excited about TQFTs :) –  Kevin H. Lin Jul 17 '10 at 19:56
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