Representation Theory of Finite Dimensional Algebras

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The investigator's main theme is to explore the representation theoryof wild finite dimensional algebras by analyzing specific classes ofrepresentations and the homomorphisms among them with an array ofmethods, ranging from geometric through combinatorial tohomological. Roughly, these classes can be grouped under thefollowing headings: (1) Classes of modules having fixed sequence ofradical layers and, more generally, classes of modules with fixed topand fixed vector space dimension. (2) Classes of modules formingcontravariantly finite or covariantly finite subcategories within thefull category of finitely generated modules. (3) Classes of infinitedirect sums (mainly of generic and finitely generated modules)enjoying finiteness conditions over their endomorphism rings. Arelated investigation aims at the geometry of finite complexes ofmodules and derived categories as an alternate setting for the studyof maps among representations. There are methodologicalinterconnections among these lines, as well as strong links in termsof the overall strategy of gaining a better understanding of algebrasthat have wild representation type. Under a wider angle, the project should be seen in the followingframework: The majority of our physical models for the universe areplaced within finite dimensional vector spaces. Frequently, suchspaces carry additional structure, such as a Lie or associativealgebra structure, for example. Associative finite dimensionalalgebras are the objects of the investigator's research. One approachtowards understanding them is to explore their representations, whichcan be thought of as 'linearized photographs' within sets of squarematrices, both coarse-grained and fine-grained pictures containingvaluable information. The representation type of the consideredalgebra is called 'tame' if the representations of any fixed matrixsize can be subdivided into finitely many 'manageable' series (theseseries themselves may be infinite), otherwise it is called 'wild'.The wild case being the one predominantly encountered in mathematicalnature, this is the case which the investigator is presentlytackling. Her idea is to thoroughly understand important subclassesof the full class of representations of algebras of wild type, and tothus establish an interlinked mosaic of analyzed territory. Attentionis being paid to choosing lines that optimally connect with the needsof adjacent fields (such as the theory of algebraic or quantum groups)and with the body of insights already established.