# Are there natural examples of non-symmetric Frobenius algebras?

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?

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.

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.

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.

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