Thursday, 24th November 2005, 4:00 pm

Mathematics Building, Lecture Hall 2

Birge Huisgen-Zimmermann

(University of California at Santa Barbara)

"Geometric aspects of the representation theory

of finite-dimensional algebras -- Overview"

** Abstract: **
This lecture is designed to be understood by a
general mathematical audience, including graduate students with some
background in basic algebraic concepts (such as would be covered in an
introductory graduate course in algebra, plus low-level familiarity
with the concept of an affine/projective algebraic variety).

We will start by discussing the ultimate classification goal in the study of a finite dimensional (associative) algebra $A$ over an algebraically closed field, illustrate this goal with some first examples, and introduce the concepts of finite, tame, and wild representation type. In particular, this will lead us to representations of algebras given by quiver and relations, which are located at the heart of this sequence of talks (we will introduce them here, no prior familiarity is required).

There will be a reception before the lecture at 3:30 outside the lecture hall.

There will be two further talks in the series, held as joint meetings of the Jerusalem Geometry/Topology Seminar and the Amitsur Algebra Seminar.

Part II: Degenerations Monday, November 28th Room 207 12.00-13.00

** Abstract:** The only additional prerequisite required for
this talk is the material of the overview lecture. Theory concerning
algebraic group actions will be explicitly quoted as needed.

This part addresses arbitrary algebras $A$ over an algebraically closed field. We will reinforce the concept of a degeneration of a finite dimensional $A$-module with initial examples. Then we will discuss the projective varieties introduced at the end of the first lecture and exploit them to advance the degeneration theory of $A$. No proofs of the theorems will be included in the lecture (but I will be available to present proofs outside the lectures to anybody who is interested). Instead, we will emphasize concrete interpretations of the theory. In particular, we will demonstrate its force in applying it to concrete examples (these can be presented graphically, without a big buildup of technical and notational ballast, with the aid of a diagrammatic method which, in essence, was introduced by Alperin).

Part III: Classification Thursday, December 1st Room 209 12.15-13.15

**Abstract:** 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.)

List of talks, 2005-06

Archive of talks