Geometric aspects of finite dimensional algebras: Uniserial representations

Abstract

K. Bongartz and B. Huisgen-Zimmermann studied uniserial modules over finite dimensional algebras. Given a path p ∉ I, they associate to it an affine variety Vp which parameterizes the uniserial modules with mast p. This variety can be calculated algorithmically. It is here shown that these varieties characterize the monomia algebras. An open problem asks for an invariant characterization of algebras isomorphic to monomial algebras. We obtain an algorithmic solution for the open problem for a class of algebras which we call "loosely constricted", and we give a necessary condition for an algebra to be isomorphic to a monomial algebra, which is algorithmic. We then give an analogous version of a theorem of Bardzell and Green which is an invariant characterization of algebras isomorphic to monomial algebras, using uniserial modules. A basis of Ext1L (U, V) is described, where the quiver has no oriented cycles and the masts of U and V pass through the same vertices, and we get an upper bound for its dimension. Here, each nonzero element of Ext1L (U, V) represents an indecomposable module. Isomorphism classes of uniserial modules over biserial algebras are described. We study alpha( U), the number of indecomposable summands of the middle term of an almost split sequence ending in U, where U is a uniserial Λ-module, and give an upper bound for it in the case that Λ is m-multiserial algebra. Irreducible radical embeddings of uniserial modules over triangular multiserial as well as monomial algebras are classified. This confirms a conjecture of A. Boldt in these cases.