Semilinear clannish algebras

Abstract: Abstract: Indecomposable modules over string algebras were classified by Butler and Ringel, and take exactly one of two forms: string modules, defined by walks in the quiver; or band modules, given by cyclic walks together with a representation of the Laurent polynomial ring. Clannish algebras, introduced by Crawley-Boevey, are a generalisation of string algebras – where one specifies a set of special loops, each bounded by some quadratic polynomial. An analogue of Butler and Ringel’s result was given where the class of string (or band) modules splits into so-called asymmetric and symmetric subclasses. Said symmetry is given by reflecting the walk about a special loop, and symmetric strings and bands are parameterised using appropriate replacements for the Laurent polynomial ring.

Both string algebras and clannish algebras were defined over a field, and the quadratics bounding special loops were assumed to have distinct roots in this field. This talk will be about ongoing joint work with Bill Crawley-Boevey, where we generalise the classification for clannish algebras. For the rings we consider we replace this field with a division ring equipped with a set of automorphisms, indexed by arrows of the quiver, and we allow irreducible quadratics to bound the special loops. The resulting notion of a semilinear clannish algebra recovers a generalisation of string algebras considered by Ringel, where one allows the map associated to an arrow to be semilinear with respect to its automorphism.

combinatoricscategory theoryrepresentation theory

Audience: researchers in the topic

( slides )

The TRAC Seminar - Théorie de Représentations et ses Applications et Connections