Throughout, we assume our algebras are basic. For a representation-finite (RF) selfinjective algebra $A$ over algebraically closed field $K$, we say $A$ is standard if $K(\Gamma_A) \simeq \mathrm{ind}-A$. Here $\Gamma_A$ is the Auslander-Retien (AR) quiver of $A$, $K(\Gamma_A)$ is the mesh category, and $\mathrm{ind}-A$ is the full subcategory of $\mathrm{mod}-A$ with objects being indecomposable (f.d.) $A$-modules. This definition is used throughout the classification of RF selfinjective algebras and related works by Riedtmann, Bongartz-Gabriel and Bretscher-Läser-Riedtmann.
However, it seems to me a different definition is used for representation-infinite (RI) selfinjective algebras. This definition, I believe, is originated from Skowronski in [Selfinjective algebras of polynomial growth. Math. Ann. 285 (1989), no. 2, 177–199]: we say a selfinjective algebra $A$ is standard, if there exists a Galois covering $R\to R/G=A$ such that (1) $R$ is simply connected locally bounded category and $G$ is an admisible torsion-free group of $K$-linear automorphisms of $R$.
A detailed description for the terminologies in this definition can be found in the same paper or even better in the survey by Skowronski in [Selfinjective algebras: finite and tame type. Trends in representation theory of algebras and related topics, 169–238, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006] It is remarked in Skowronski's paper that such notion of standard is the same as the one described in the first paragraph, when $A$ is representation-finite. My question is:
When $A$ is RI selfinjective and standard, do we have $K(\Gamma_A)\simeq \mathrm{ind}-A$? In fact, it seems to me the new definition is implicitly saying that "$K(\Gamma_A)\simeq \mathrm{ind}-A$" does not make sense, am I correct or wrong? If so, what's wrong with such statement in RI selfinjective algebras?
