References request: Auslander-Reiten theory of algebras like $B_{k,n}$

Question

In the paper A categorification of Grassmannian cluster algebras, an algebra $B_{k,n}$ is defined as follows.

Denote by $C=(C_0, C_1)$ the circular graph with vertex set $C_0=\mathbb{Z}_{n}$ clockwise around the circle, and with the edge set $C_1=\mathbb{Z}_n$, with edge $i$ joining vertices $i-1$ and $i$, see Figure 3 on page 8 of the paper.

Denote by $Q_C$ the quiver with the same vertex set $C_0$ and with arrows $x_i: i-1 \to i$, $y_i: i \to i-1$ for every $i \in C_0$, see Figure 3 on page 8 of the paper.

The algebra $B_{k,n}$ (it is denoted by $\overline{A}$ in the paper, see the paragraph before Remark 3.4 in the paper) is the quotient of the complete path algebra $\widehat{\mathbb{C} Q_C}$ by the ideal generated by the $2n$ relations $x y = y x$, $x^{k} = y^{n-k}$ (two relations for each vertex of $Q_C$), where $x, y$ are arrows of the form $x_i, y_j$ for appropriate $i,j$.

Is the algebra $B_{k,n}$ an Artin algebra? If it is not an Artin algebra, has the Auslander-Reiten theory developed for this type of algebras? I am asking this question because I found that Auslander-Reiten theory is used in the paper and I would like to know some references. Thank you very much.

Answer 1

I think the algebra you describe is actually the algebra denoted by $\hat{A}$ in the paper you reference. $\bar{A}$ is an uncompleted version.

Neither algebra is an Artin algebra, or even an artinian ring, since the powers of the ideal generated by the arrows form an infinite decreasing sequence of ideals.

As far as I can see, the paper only uses the existence of Auslander-Reiten sequences for the category of (maximal) Cohen-Macaulay $\hat{A}$-modules, and justify their existence by a reference in Remark 3.3 to Auslander's paper:

Auslander, Maurice, Isolated singularities and existence of almost split sequences. Notes by Louise Unger, Representation theory II, Groups and orders, Proc. 4th Int. Conf., Ottawa/Can. 1984, Lect. Notes Math. 1178, 194-242 (1986); identical with: Representations of algebras, Proc. 4th Int. Conf., Ottawa/Can. 1984, Vol. 1, Carleton-Ottawa Math. Lect. Note Ser. 1, W5, 49 p. (1985). ZBL0633.13007.