Noncommutative algebra

2024/2025

Programme:

Financial Mathematics, Second cycle

Year:

1 ali 2 year

Semester:

first or second

Kind:

optional

Group:

ECTS:

Language:

slovenian, english

Course director:

Acad. Prof. Dr. Matej Brešar, Prof. Dr. Jakob Cimprič, Prof. Dr. Igor Klep

Hours per week – 1. or 2. semester:

Lectures

Seminar

Tutorial

Lab

There are no prerequisites.

Noncommutative division rings. Frobenius' theorem. Wedderburn's theorem on finite divison rings.
Radical. Semisimple algebras. Wedderburn's theorem. Maschke's theorem.
Simple and semisimple modules. Density theorem. Jacobson radical.
Tensor product of algebras. Skolem-Noether theorem. Double centralizer theorem. Brauer group.

M. Brešar: Introduction to noncommutative algebra, Cham : Springer, cop. 2014.
B. Farb, R. K. Dennis: Noncommutative algebra, New York : Springer, cop. 1993.
T. Y. Lam: A first course in noncommutative rings, 2nd ed., New York : Springer, cop. 2001.
R. S. Pierce: Associative algebras, New York : Springer, cop. 1982.
L. H. Rowen: Graduate algebra : noncommutative view, Providence : American Mathematical Society, cop. 2008.

To master basic concepts and tools of noncommutative algebra.

Knowledge and understanding:
Understanding of basic concepts and theorems of noncommutative algebra, and their role in some other areas.
Application:
In other mathematical areas.
Reflection:
Understanding the theory on the basis of examples and applications.
Transferable skills:
Formulation and solution of problems using abstract methods.

Lectures, exercises, homeworks, consultations.

Homeworks
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Matej Brešar:
BREŠAR, Matej, CHEBOTAR, M. A., MARTINDALE, Wallace S. Functional identities, (Frontiers in mathematics). Basel, Boston, Berlin: Birkhäuser, cop. 2007. XII, 272 str. ISBN 978-3-7643-7795-3. ISBN 978-3-7643-7796-0. [COBISS-SI-ID 14332505]
BREŠAR, Matej. An elementary approach to Wedderburn's structure theory. Expositiones mathematicae, ISSN 0723-0869, 2010, vol. 28, no 1, str. 79-83. [COBISS-SI-ID 15382617]
BREŠAR, Matej. An alternative approach to the structure theory of PI-rings. Expositiones mathematicae, ISSN 0723-0869, 2011, vol. 29, no 1, str. 159-164. [COBISS-SI-ID 15859545]
Jaka Cimprič:
CIMPRIČ, Jaka. Free skew fields have many [ast]-orderings. Journal of algebra, ISSN 0021-8693, 2004, vol. 280, no. 1, str. 20-28. [COBISS-SI-ID 13210201]
CIMPRIČ, Jaka. Formally real involutions on central simple algebras. Communications in algebra, ISSN 0092-7872, 2008, vol. 36, no. 1, str. 165-178. [COBISS-SI-ID 14613337]
CIMPRIČ, Jaka, HELTON, J. William, MCCULLOUGH, Scott, NELSON, Christopher. A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: algorithms. Proceedings of the London Mathematical Society, ISSN 0024-6115, 2013, vol. 106, iss. 5, str. 1060-1086. [COBISS-SI-ID 16636249]
Igor Klep:
Derksen, Harm, KLEP, Igor, Makam, Visu, Volčič, Jurij. Ranks of linear matrix pencils separate simultaneous similarity orbits. Advances in mathematics. - ISSN 0001-8708 (Vol. 415, Feb. 2023, art. 108888 (20 str.)) [COBISS-SI-ID 141160195]
Helton, J. William, KLEP, Igor, Volčič, Jurij. Factorization of noncommutative polynomials and Nullstellensätze for the free algebra. International mathematics research notices. - ISSN 1073-7928 (Vol. 2022, iss. 1, Jan. 2022, str. 343-372) [COBISS-SI-ID 48375811]
KLEP, Igor, Pascoe, James E., Volčič, Jurij. Positive univariate trace polynomials. Journal of algebra. - ISSN 0021-8693 (Vol. 579, Aug. 2021, str. 303-317) [COBISS-SI-ID 105932803]