Finite, Tame and Wild Symmetric Special Multiserial Algebras
Algebra and Geometry Seminar
11th October 2017, 2:30 pm – 3:30 pm
Howard House, 4th Floor Seminar Room
The representation type of an algebra is of tremendous importance to representation theorists. It essentially describes how complicated it is to understand the representation theory of an algebra. Drozd’s famous trichotomy states that an algebra is of finite, tame or wild representation type – and these are mutually exclusive. On the simplest end on the spectrum are algebras of finite representation type, which have finitely many indecomposable modules over the algebra. On the opposite end of the spectrum are the algebras of wild representation type, where the representation theory is at least as complicated as the representation theory of all finite dimensional algebras. Thus, a classification of the indecomposable modules over a wild algebra is often considered hopeless. In the middle are the algebras of tame representation type, where there remains some hope of classifying the indecomposable modules, and thus, these are of great interest to representation theorists.
Special biserial algebras are a class of algebras that have been of immense interest and study within the representation theory of finite dimensional algebras. They are defined by quiver and relations and have a highly combinatorial structure. Special biserial algebras are of tame representation type, and their indecomposable modules have been classified. Recently, there has been a lot of interest in a generalisation of these algebras – namely, the class of special multiserial algebras, which were first defined by Von Höhne and Waschbüsch, and later studied by Green and Schroll. Unlike special biserial algebras, most (but not all) special multiserial algebras are wild. We are particularly interested in symmetric special multiserial algebras, since these are associated to a decorated hypergraph (with orientation) called a Brauer configuration. Brauer configurations are incredibly useful since the representation theory of the associated algebra is encoded in the combinatorial data of the hypergraph. In this talk, we will provide a complete classification of all finite, tame and wild symmetric special multiserial algebras. Time permitting, we will provide further details on the structure of tame symmetric special multiserial algebras.