In their article https://www.sciencedirect.com/science/article/pii/0001870891900378 , Auslander and Reiten suggested a generalisation of Gorenstein algebras for Artin algbras which they call Cohen-Macaulay (=CM) Artin algebras. By definition an Artin algebra $A$ is CM iff there are adjoint functors $F$ and $G$ between the subcategories of mod-A consisting of modules of finite projective dimension and finite injective dimension. We can write $F=Hom_A(w,-)$ for an $A$-bimodule $w$ (which must be the strong cotilting module).

It seems that since then there were no other articles related to this subject (besides a survey article by Auslander and Reiten one year later). I think the reason might be that no non-trivial example of CM Artin algebras were known besides Gorenstein algebras and finitistic dimension 0 algebras (and a tensor product construction by tensoring a CM algebra with the algebra of upper triangular 2x2 matrices) which might be called trivial examples.

Despite searching with the computer for a long time, I never found a non-trivial example but it seems the computer found now such an example using QPA.

Namely let $A$ be the quiver algebra with quiver Quiver( ["v1","v2","v3"], [["v1","v2","a1"],["v2","v1","a2"],["v2","v3","a3"],["v3","v2","a4"]] )

and relations: [ a2*a1-a3*a4, a4*a3, a2*a1*a3,a4*a2*a1, a1*a2*a1*a2,a2*a1*a2*a1 ].

Then $A$ should be CM (with infinite Gorenstein dimension) with strong cotilting module $w=eA \oplus \Omega^2(D(A))$ when $eA$ is the minimal faithful projective-injective $A$-module (note that $A$ has dominant dimension equal to two).

Question 1: Is this algebra really CM? (the computer can not test it for sure, but I think it should be correct in case I understand the results of Auslander and Reiten correctly).

Note that the algebra is not of any "trivial form": It is not Gorenstein, has finitistic dimension 2 and its dimension is 13 which is not divisble by 3 (so it is not obtained by tensoring with the algebra of upper triangular 2x2 matrices). As a sidequestion it might be interesting to know whether this is really the first "nontrivial" examples or any example of CM Artin algebras outside the articles of Auslander and Reiten.

Question 2: It is an open question by Auslander and Reiten whether for a CM Artin algebra $A$ with strong cotilting module $w$ the trivial extension $B$ of $A$ by $w$ is Gorenstein. What is the algebra $B$ in this case?

(Note that one should have $End_A(w) \cong A$, giving $w$ a bimodule structure. But I can hardly imagine this in practise so I do not find a good way to calculate with the bimodule structure in order to obtain $B$)

Question 3: It is another open question by Auslander and Reiten whether for CM Artin algebras $A$ and $B$, one has that also $A \otimes B$ is CM. Is in this example the enveloping algebra $A \otimes A$ CM?

edit: Some additional information on A. It has 20 indecomposable modules (thus it is reprentation-finite), it has Loewy length 4 and it has 6 indecomposable modules of finite projective (injective) dimension.