Are there natural examples of non-symmetric Frobenius algebras?

Question

Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in FVect, or perhaps in some more exotic category?

Answer 1

Here are a couple of "natural" constructions that produce Frobenius algebras over a field that are not necessarily symmetric.

(A) The trivial extension algebra of any algebra $A$ is defined to be $A\oplus DA$, where $DA$ is the vector space dual of $A$, with multiplication $(a,\phi)(b,\theta)=(ab,a\theta+\phi b)$ using the natural $A$-bimodule structure on $DA$. This is a symmetric algebra, but if you twist the action of $A$ on $DA$ on one side by an algebra automorphism of $A$, then you get a Frobenius algebra which is not in general symmetric.

(B) If $A$ is a Frobenius algebra with an action of a finite group $G$, then the skew group algebra $A\ast G$ is a Frobenius algebra which is not necessarily symmetric, even if $A$ is.

Answer 2

Though "natural" isn't a mathematical term with a strict meaning here, it's worth considering the work of Larson and Sweedler on finite dimensional Hopf algebras in Amer. J. Math. 91 (1969), 75-94. One gets many examples of such Hopf algebras by forming the restricted enveloping algebra of a finite dimensional restricted Lie algebra over a field of prime characteristic. These are always Hopf algebras, therefore automatically Frobenius algebras (Larson-Sweedler), but they may fail to be "unimodular" and thus fail to be "symmetric" in the sense of Nesbitt. This theme was further refined in a note of mine here.

[EDIT] An example mentioned by Larson-Sweedler is itself fairly natural: the unique up to isomorphism nonabelian (not nilpotent, but solvable) 2-dimensional Lie algebra over a field, with a basis $x,y$ and nonzero commutator $[xy] = x$. In characteristic $p>0$ this yields a restricted Lie algebra structure with $x^{[p]} = 0$ and $y^{[p]} = y$. Its restricted enveloping algebra is a Hopf algebra of dimension $p^2$, hence Frobenius but not unimodular and thus not symmetric.

On the other hand, it's certainly true that group algebras of finite groups and restricted enveloping algebras of many well-behaved Lie algebras do turn out to be symmetric. However, it's often rather subtle to prove this in cases involving Lie algebras and quantum groups.

Answer 3

Another construction is the preprojective algebra of a Dynkin quiver (a quiver such that the underlying graph is Dynkin).

Such an algebra is Frobenius but not in general symmetric, see for instance Erdmann, Karin; Snashall, Nicole, On Hochschild cohomology of preprojective algebras. I.

Answer 4

While symmetric Frobenius algebras in symmetric monoidal categories play a role in 2D TFT, Frobenius algebras in symmetric monoidal categories with "Nakayama automorphism" which is an involution play a role for TQFT on spin surfaces. The symmetric ones (i.e. trivial Nakayama automorphism") give exactly TFTs not depending on the spin structure, see http://arxiv.org/abs/1402.2839