Non-Commutative Algebra and Geometry
(Fall 2002)
Course Description.
The official course description
is a reasonably honest account of what we will do, but
my motivation for teaching this course
is coming from string theory. I know essentially no physics. But
recent string theory papers are filled with various non-commutative
algebras that are ``old friends'' of mine. I would like to address some
questions about these algebras that are of interest to
string theorists. There is a language barrier unfortunately.
All I know for sure is that the string theorists are interested
in the finite-dimensional representations of (= modules over) these
algebras. What these algebras have in common is that they are
finite modules over their centers, and their centers correspond to
interesting algebraic varieties; these varieties are usually
singular and the string theorists are interested in the
non-commutative algebras because they provide some sort of
``non-commutative resolutions'' of the varieties. The ``points'' on
the non-commutative space are provided by the finite-dimensional
simple modules over the non-commutative algebras.
I am sure that some of you will
know much more physics than I do. I will not attempt to explain any
physics in this course, but here are some papers by
string theorists in which you
will find all sorts of interesting non-commutative algebras:
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D. Berenstein, V. Jejjala, and R.G. Leigh,
Marginal and Relevant Deformations of N=4 Field Theories and
Non-Commutative Moduli Spaces of Vacua
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D. Berenstein, V. Jejjala, and R.G. Leigh,
Non-Commutative Moduli Spaces, Dielectric Tori and T-duality
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D. Berenstein and R.G. Leigh,
Non-Commutative Calabi-Yau Manifolds
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D. Berenstein and R.G. Leigh,
Resolution of Stringy Singularities by Non-commutative Algebras
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D. Berenstein,
On the universality class of the conifold
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D. Berenstein,
Reverse geometric engineering of singularities
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M.R. Douglas and G. Moore,
D-branes, Quivers, and ALE Instantons
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B. Feng, A, Hanany and Y-H He,
D-Brane Gauge Theories from Toric Singularities and Toric Duality
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H. Kim and C-Y Lee,
Noncommutative K3 Surfaces
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N.A. Nekrasov,
Lectures on open strings, and noncommutative gauge fields
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N. Seiberg and E. Witten,
String Theory and Noncommutative Geometry
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A.M. Uranga,
From quiver diagrams to particle physics
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A.M. Uranga,
Brane Configurations for Branes at Conifolds
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E. Witten,
Noncommutative Geometry and String Field Theory,
Nucl. Phys., B268 (1986) 253.
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And here's a book:
Physics in Noncommutative World: I Field Theories
Ed by Miao Li and Yong-Shi Wu.
This course will cover some of the algebraic background needed to understand the algebras
appearing in these string theory papers. There will be no physics in this course.
Course Notes.
These are posted in three file formats: pdf, dvi, and postscript.
Chapter 1. Rings, Modules, and Algebras. This is background but worth
reading anyway---think of it as a refresher course. And here are the dvi
and ps files.
Chapter 2. Noncommutative Polynomial Equations.
And here is the ps file.
Chapter 3. Prime ideals and Spec(A).
And here is the ps file.
Chapter 4. Quick Homological Algebra.
And here are the ps and dvi files.
Chapter 5. Artinian Rings.
And here is the ps file.
Chapter 6. Representations of Quivers.
And here is the ps file.
As you will have guessed, I have fallen way behind in posting course notes. I hope to catch
up with what we have covered in class... and when I do I will post more course notes.
If you are following this course you might also be interested in
visiting Lieven LeBruyn's noncommutative algebra & geometry pages,
na&g,
for short, and also
his course page where you will find notes
for some courses he is teaching.