Noncommutative Algebra

Recommended Prerequisites

Concepts from abstract Algebra and Linear Algebra.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required the solution of exercises, course projects or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Noncommutative Algebra at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Algebra, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

The syllabus will vary from year to year but the core syllabus is

- Division rings: Quaternions, Cayley-Dickson construction

- Wedderburn-Artin Theory: matrix rings over division rings and

semisimple modules

- Tensor products and categories

- Algebras presented by generators and relations

- Simple algebras: Weyl algebras, central simple algebras, Brauer group

Further topics (an uncomplete list):

- Non-commutative polynomial rings: skew-polynomial rings, differential operator algebras

- Lie algebras and their enveloping algebras

- basic facts on Hopf algebras: Frobenius algebras, Maschke theorem, Noetherian Hopf algebras

- Noetherian ring theory: non-commutative localization and Goldie's theorem

- basic facts of homological algebra.

Head Lecturer(s)

Christian Edgar Lomp

Assessment Methods

Assessment
Other: 10.0%
Frequency: 40.0%
Exam: 50.0%

Bibliography

C. Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995.

M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series W.A. Benjamin, Inc., New York, 1969.

S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993.

J. J. Rotman, An Introduction to Homological Algebra, 2nd edition, Universitext, Springer, New York, 2009.

T.Y. Lam, A First Course in Noncommutative Rings, 2nd edition, Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001.

B. Farb and R.K. Dennis, Noncommutative Algebra, Graduate Texts in Mathematics, vol. 144. Springer-Verlag, New York, 1993.

K.R. Goodearl and R.B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004.

D.S. Passman, A Course in Ring Theory, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991.

J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings. With the cooperation of L. W. Small, Revised edition. Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001.

K. Erdmann and M. J. Wildon, Introduction to Lie algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2006.