Semilinear clannish algebras
String algebras are monomial algebras introduced by Butler and Ringel, where they showed any indecomposable representation is: a string module, given by a relation-avoiding walk in the quiver; or a band module, given by a cyclic walk and some module over the Laurent polynomial ring. Clannish algebras, introduced by Crawley-Boevey, generalise string algebras - in addition to monomial relations, one specifies a set of special loops, each bounded by some monic quadratic polynomial. Butler and Ringel’s classification was then adapted, where the class of string (or band) modules splits into asymmetric and symmetric subclasses. Said symmetry is a reflection of the walk about a special loop, and symmetric strings and bands are parameterised by replacements for the Laurent polynomial ring.
Both string algebras and clannish algebras are defined over a field, and the quadratics bounding special loops must factor with distinct roots in this field. This talk is based on joint work with Crawley-Boevey (2204.12138), where we generalise the module classification for clannish algebras. We replace the ground field with a division ring, we equip each arrow with an automorphism of this division ring, and we allow irreducible quadratics to bound the special loops. The resulting notion of a semilinear clannish algebra specifies to a generalisation of string algebras considered by Ringel, where the map associated to an arrow in any representation must be semilinear with respect to its automorphism.