Stable equivalence closure for Brauer graph algebras beyond symmetric targets

Determine whether a Brauer graph algebra, excluding algebras with radical square zero, can be stably equivalent to a non-symmetric algebra.

Background

The paper studies whether the class of Brauer graph algebras is preserved under stable equivalence. It notes that the question is unresolved in general and that a negative answer is possible, since the Brauer graph algebra k[x]/(x2) is stably equivalent to the hereditary algebra k(·→·). The authors then identify the more specific unresolved issue of whether the obstruction persists after excluding algebras of radical square zero.

The paper proves preservation under additional hypotheses, including stable equivalence with a basic symmetric target and stable equivalence of Morita type. These results do not settle the unrestricted question involving potentially non-symmetric target algebras.

References

However, if we exclude the algebras of radical square zero, then we even do not know whether a Brauer graph algebra can be stably equivalent to some non-symmetric algebra.

— Brauer graph algebras are closed under stable equivalence of Morita type  (2608.14253 - Chen et al., 14 Aug 2026) in Section 1, Introduction

However, we even do not know a concrete example where a symmetric algebra is stably equivalent to some non-symmetric algebra over an algebraically closed field, although there do exist such examples over a non-algebraically closed field (see ).

— Brauer graph algebras are closed under stable equivalence of Morita type  (2608.14253 - Chen et al., 14 Aug 2026) in Section 2.4, subsection “Stable equivalences of Morita type and centers”

The question of whether Brauer graph algebras are closed under stable equivalence remains open.

— Classification of Brauer graph algebras under stable equivalence of Morita type  (2608.22715 - Li et al., 24 Aug 2026) in Section 1, Introduction