# Course Details

#### Name:

**Advanced Algebra 2 / Algebra Superiore 2**

### Basic information

##### Credits:

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

##### Degree(s):

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

##### Language:

English

### Course Objectives

To provide a good understanding of the theory of Lie algebras giving a particular emphasys to the classification of simple Lie algebras.

### Course Content

- 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.

### Prerequisites and Learning Activities

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

Lectures to be attended in class

### Assessment Methods and Criteria

Oral examination

### Textbooks

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

### Course page updates

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

*Course information last updated on: 11 marzo 2015, 14:59*