Fall 2019 - Section 1

Schedule: Mo/We 10.45 AM - 12.00 PM ENR2 S395.
**Instructor:** K. Lux

- Office: room S335 ENR2
- Office Hours: by appointment.

**Text:**
''Associative Algebras '' by R.S. Pierce.

See also:

- Finite Dimensional Algebras by Y.A. Drozd and V.V.Kirichenko.
- Representations and cohomology by D.J. Benson.
- Input for GAP as demonstrated in class on 9/11/2019.
- a paper by Ed Green explaining noncommutative Groebner basis techniques for path algebras.

For more information about quaternion algebras in particular their history see the book Quaternion Algebras by J. Voight .

For more material on Groebner bases and examples, see

The notion of an algebra over a field goes back to the beginning of the 20th century and generalizes the notion of a noncommutative ring. Algebras arise in several different areas for example group theory, number theory, and differential geometry. Examples range from matrix algebras, group algebras, endomorphism algebras, division algebras, to (quantum) universal enveloping algebras.

The most important way to study an algebra is via its homomorphisms into matrix algebras also called the (matrix) representations of the algebra. Representation theory of algebras turns out to be of essential use in such diverse topics as the representation theory of groups, the representation theory of Lie algebras, and the study of the Brauer group of a field in algebraic number theory. In differential geometry the representations of Frobenius algebras play a central role in the classification of 2-dimensional topological quantum field theories and the representations of quiver algebras arise in string theory and in algebraic topology, in the form of persistent homology..

The goal of this special topics course is to give an introduction to the representation theory of algebras following mainly the Graduate Text in Mathematics (nr. 88), ”Associative Algebras” by Richard S. Pierce, published by Springer (the electronic version is available for free).

We will give an overview over the structure theory for algebras, in particular the theory of their indecomposable modules and their projective modules. This will lead us to the notion of a quiver algebra and quiver reprsetations. and we will look at the theoretical and practical appplications of quiver representations. Additional topics that we will try to highlight is the (Hochschild) cohomology of algebras, the theory of Brauer groups, and computational aspects such as noncommutative Groebner basis algorithms. This will include hands-on experience by working with algebras in the computer algebra system GAP amd the GAP-package QPA (quivers and path algebras).

This special topics course is addressed to students interested in algebra, group theory, number theory, mathematical physics, and differential geometry and can also serve as an introduction to one of my research areas. As a prerequisite for the course the material covered in the first year core course in abstract algebra should suffice.

K. Lux / Department of Mathematics / University of Arizona / last revised Septembert 2019At the end of the semester, students will present a talk on a topic of their choice related to the material covered in the course. Potential topics will be posted later during the semester.

**Homework:**
Homework will be assigned
regularly.