Non-commutative Algebra

Code:

M561

Acronym:

M561

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	3	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

The aim of this course is to introduce the students to some aspect of non-commutative (associative) algebras and their modules.

Learning outcomes and competences

The expected outcome is that the student knows some of the classicsal results of non-commutative algebra.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra and basic abstract algebra, including basic facts on groups, rings and modules.

Program

The program of this course will focus on the structure of non-commutative rings. We will cover basic facts on division rings and quaternion algebras. The construction of the Brauer group of a field and its basic properties will be discussed. A brief outlook to the Brauer group of a commutative ring will be provided. If time permits, we will discuss Noetherian rings and Goldie's theorem, skew polynomial rings and division rings that are infinite dimensional over their center and certain classes of Hopf algebras.

Mandatory literature

H.J. Schneider; Lectures on Hopf algebras, 1994 (http://www.famaf.unc.edu.ar/andrus/papers/Schn1.pdf)
Sweedler Moss E.; Hopf algebras. ISBN: 8053-9255-6
Susan Montgomery; Hopf algebras and their actions on rings, AMS, 1993. ISBN: 978-0-8218-0738-5 (https://link.springer.com/book/10.1007/978-0-387-72766-0)

Complementary Bibliography

Dascalescu Sorin; Hopf algebras. ISBN: 0-8247-0481-9
Abe Eiichi; Hopf algebras. ISBN: 0-521-22240-0
Caenepeel Stefaan; Brauer groups, Hopf algebras and Galois theory. ISBN: 0-7923-4829-X

Teaching methods and learning activities

Traditional teaching

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Frequência das aulas	100,00
Total:	100,00

Eligibility for exams

Whether the student participates in class is up to him or her and has no influence on the final grade.

Calculation formula of final grade

The final grade is the sum of the midterm quiz and the final exam.

Observations

Classes will be given at the university of Coimbra.