# Representation Theory – Unquarted quiver algebra equivalent to its opposite algebra

Let $$A = KQ / I$$ to be a quiver algebra with the following two properties:

a) Q is an acyclic quiver

(b) the injective envelope of $$A$$ is projective.

Question 1: Is there an algebra $$A$$ with properties a) and b) such that $$A$$ is not derived equivalent to its opposite algebra $$A ^ {op}$$?

(In case you know of simple examples, it would also be interesting to see an algebra $$A$$ non derivative equivalent to $$A ^ {op}$$ with just the goods a) or b))

Question 2: Does each algebra of Nakayama $$A$$ derivative derived from $$A ^ {op}$$?

Note that at https://www.sciencedirect.com/science/article/pii/S0021869315000575, at least the singularity categories are equivalent.