Algebras of generalized quaternion type are self-injective algebras whose stable Auslander-Reiten quiver contains components of type ZD-infinity — the infinite dihedral quiver. These algebras generalize the group algebras of generalized quaternion groups, which are the simplest examples, but the class is much larger.
Skowronski and Yamagata classify the biregular case — where the algebra has a biregular Nakayama automorphism (one that permutes the simple modules in a biregular pattern). The classification produces an explicit list of algebras, defined by quivers with relations, that exhausts all possibilities.
The biregularity condition is technical but powerful: it forces the algebra's module category to have a specific periodic structure that constrains the possible quivers and relations. The classification proceeds by analyzing the socle deformation theory — how the algebra's socle (minimal ideal structure) interacts with the Nakayama automorphism.
The result connects representation theory to topology through the quaternion group's role as the fundamental group of certain 3-manifolds. The group algebras of generalized quaternion groups control the periodic projective resolutions that appear in the topology of 3-dimensional lens spaces, and the classification of algebras of generalized quaternion type extends this topological connection to a broader algebraic class.
A classification of algebras sharing the module-theoretic DNA of quaternion groups. The quaternion structure — periodic, self-injective, dihedral — propagates from groups through their algebras to a full family.