Given a nonsemisimple symmetric algebra B and a non-selfinjective algebra A (all algebras are finite dimensional over a field and connected).
Can A and B have isomorphic Hochschild-cohomology rings?



A question on Hochschild cohomology
There are examples of non-semisimple algebras with trivial Hochschild cohomology: for example, the path algebra $C$ of a quiver whose underlying graph is a tree.


Also, for finite dimensional algebras, the Hochschild cohomology algebra of a tensor product of algebras is the tensor product of their Hochschild cohomology algebras.


So take any symmetric algebra $B$, and let $A=B\otimes\_kC$ for $C$ as above.


