Can finite dimensional algebras (over a field $K$) A with $D(A) \otimes\_A D(A) \cong A$ as $A$-bimodules be somehow classified? I think taking for A a selfinjective algebra with nakayama permutation of order at most two should work. Are there nonselfinjective examples? Here $D(A)=Hom\_K(A,K)$ is the dual of the regular module.
Here a little motivation:


Symmetric algebras are characterised by $D(A) \cong A$ as bimodules.


More generally define the cyclic monoid generated by $D(A)$ with multiplication $\otimes\_A$, when is this a group? That is, which algebras satisfy $A \cong D(A)^{\otimes i}$ for some $i \geq 1$? 
Are there non-selfinjective examples? 
It might also be interesting to look just at one-sided isomorphisms instead of bimodule isomorphisms, where this notion might generalise frobenius algebras.



$D(A) \otimes D(A)= A$?
If $A\cong D(A)^{\otimes i}$as $A$-bimodules, for some $i\geq1$, then $-\otimes\_AD(A)$ is a self-equivalence of the module category, and so takes projectives to projectives. But it takes $A$ to $A\otimes\_AD(A)\cong D(A)$, so $A$ must be self-injective.


