The main goal of representation theory is to understand all representations of a given algebra and their homomorphisms. It is enough to classify the indecomposable representations and their homomorphisms for this. However, for most algebras, this seems to be a hopeless task. Because of that, we would like to simplify the problem and focus on understanding just the generic representations. To do so we need to parametrise the representations with a variety and determine its irreducible components. In this thesis, we determine the components for varieties of representations of a class of finite dimensional algebras over an algebraically closed field, that contains all algebras of the form $k\langle x_1,\dots,x_n\rangle/((x_1,\dots,x_n)^3+(\sum_i x_i^2))$. The irreducible components in this case are determined by dimensions of radicals and socles of their generic elements. To show this we first find a cover of the variety of representations using such irreducible subvarieties and then filter out the ones that are not maximal.
|
