# Course Details

#### Name:

**Advanced Algebra / Algebra Superiore**

### Basic information

##### Credits:

*Master Degree in Mathematics:* 6 Ects (b)+6 Ects (c)

##### Term:

*Module Advanced Algebra 1:* 1° semester

*Module Advanced Algebra 2:* 2° semester

##### Degree(s):

Compulsory 1^{st} year Master Degree in Mathematics curriculum Generale

##### Language:

English

### Course Objectives

**Module Advanced Algebra 1:** Commutative algebra

**Module Advanced Algebra 2:** To provide a good understanding of the theory of Lie algebras giving a particular emphasys to the classification of simple Lie algebras.

### Course Content

##### Module Advanced Algebra 1

- Rings and ideals
- Nilradical
- Modules
- Tensor product
- Fraction rings and modules
- Primary decomposition
- Integral dependence
- Valuations
- Chain conditions
- Noetherian rings
- Artinian rings
- Dedekind rings and DVR

##### Module Advanced Algebra 2

- Abstract:
The course covers the basics of the theory of Lie algebras in characteristic 0 and their representations.
- Extended program:
Lie algebras, linear Lie algebras, derivation algebras. Ideals, homomorphisms, representations. Automorphisms. Solvability and nilpotence, Engel's theorem. Semisimple Lie algebras. Theorems of Lie and Cartan, Killing's form, complete reducibility. Representations of sl(2,F). Decomposition and root spaces. Axiomatic of root spaces. The Weil group. Classification of root spaces. Construction and abstract theory. Isomorphism Theorems. Cartan subalgebras. Conjugacy theorems. Universal enveloping algebras and the theorem of Poincaré Birkoff Witt. Simple algebras. Basic theory of the representations of Lie algebras.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

##### Module Advanced Algebra 1

- At the end of the course the student should be able to know perfectly the basics on ring theory and well the main concepts of commutative algebra up to Dedekind rings

### Prerequisites and Learning Activities

**Module Advanced Algebra 1:** Algebra 1 and 2 (now Algebra 12 credits)

**Module Advanced Algebra 2:** A basic knowledge of algebra taught in the first cycle, i.e. groups, rings, fields, polynomials.
Knowledge of linear algebra (vector spaces and matrix algebra)

### Teaching Methods

**Language**: English

**Module Advanced Algebra 1:** Theory, exercises, quizzes

**Module Advanced Algebra 2:** Lectures to be attended in class

### Assessment Methods and Criteria

**Module Advanced Algebra 1:** Oral

**Module Advanced Algebra 2:** Oral examination

### Textbooks

##### Module Advanced Algebra 1

- M.F. Atiyah/ I.G. Macdonald,
**introduzione all'algebra commutativa**. Feltrinelli. * *

##### Module Advanced Algebra 2

- James E. Humphreys,
**Introduction to Lie Algebras and Representation Theory**. Springer. (vol. GTM no. 9) 1978. *Second Edition *

### Online Teaching Resources

### Recent teaching material

This list contains only the latest published resources. Resources marked with an asterisk belong to other courses (indicated between brackets)

** Click here to access the complete resources list.**

### Course page updates

This course page is available (with possible updates) also for the following academic years:

*Course information last updated on: 21 settembre 2016, 14:59*