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Lecture 3: Geometric Aspects of the Representation Theory of Finite Dimensional Algebras-Classification | Einstein Institute of Mathematics

Lecture 3: Geometric Aspects of the Representation Theory of Finite Dimensional Algebras-Classification

Date: 
Thu, 01/12/200512:15
Location: 
Room 209, Math Building
Lecturer: 
Birge Huisgen-Zimmermann, University of California at Santa Barbara
We explain -- first in an intuitive fashion, then more formally -- what it means for a collection of finite dimensional representations to be classifiable by way of a fine or coarse moduli space (in the sense of Mumford). Then we will answer the following question, which allows to illustrate the techniques we employ in the most transparent format: When do the representations with fixed dimension and fixed squarefree top permit such a fine or coarse classification? (Given a finite dimensional algebra $A$ with Jacobson radical $J$, the top of a left $A$-module $M$ is the semisimple module $M/JM$; that it be squarefree means that no simple summand occurs with a multiplicity larger than 1). In case the answer is positive, we describe the corresponding moduli spaces, sketch (in non-technical terms) how they can be accessed by way of quiver and relations of $A$, and again give examples. (An alternate approach to guaranteeing and constructing moduli spaces for quiver representations, due to King, will be touched on the side.)