Representation Theory of Finite Dimensional Algebras

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The investigator's main theme is to explore the representation theory of wild finite dimensional algebras by analyzing specific classes of representations and the homomorphisms among them with an array of methods, ranging from geometric through combinatorial to homological. Roughly, these classes can be grouped under the following headings: (1) Classes of modules having fixed sequence of radical layers and, more generally, classes of modules with fixed top and fixed vector space dimension. (2) Classes of modules forming contravariantly finite or covariantly finite subcategories within the full category of finitely generated modules. (3) Classes of infinite direct sums (mainly of generic and finitely generated modules) enjoying finiteness conditions over their endomorphism rings. A related investigation aims at the geometry of finite complexes of modules and derived categories as an alternate setting for the study of maps among representations. There are methodological interconnections among these lines, as well as strong links in terms of the overall strategy of gaining a better understanding of algebras that have wild representation type. Under a wider angle, the project should be seen in the following framework: The majority of our physical models for the universe are placed within finite dimensional vector spaces. Frequently, such spaces carry additional structure, such as a Lie or associative algebra structure, for example. Associative finite dimensional algebras are the objects of the investigator's research. One approach towards understanding them is to explore their representations, which can be thought of as 'linearized photographs' within sets of square matrices, both coarse-grained and fine-grained pictures containing valuable information. The representation type of the considered algebra is called 'tame' if the representations of any fixed matrix size can be subdivided into finitely many 'manageable' series (these series themselves may be infinite), otherwise it is called 'wild'. The wild case being the one predominantly encountered in mathematical nature, this is the case which the investigator is presently tackling. Her idea is to thoroughly understand important subclasses of the full class of representations of algebras of wild type, and to thus establish an interlinked mosaic of analyzed territory. Attention is being paid to choosing lines that optimally connect with the needs of adjacent fields (such as the theory of algebraic or quantum groups) and with the body of insights already established.