MT016 Lie Algebras

6 ECTS - 3-0 Duration (T+A)- . Semester- 3 National Credit

Information

Code MT016
Name Lie Algebras
Semester . Semester
Duration (T+A) 3-0 (T-A) (17 Week)
ECTS 6 ECTS
National Credit 3 National Credit
Teaching Language Türkçe
Level Yüksek Lisans Dersi
Type Normal
Mode of study Yüz Yüze Öğretim
Catalog Information Coordinator Doç. Dr. ZEYNEP ÖZKURT


Course Goal

The aim of this course is to make students understand the structure of Lie algebras and to use them in applications.

Course Content

Representation of Lie algebras, Structure of Sl (n, C) Lie algebras, Cartan criterion, Root space decomposition of Lie algebras, root systems, Classical Lie algebras

Course Precondition

no

Resources

Karin Erdman, Mark Wildon Introduction to Lie Algebras

Notes

Jacobson, Lie Algebras


Course Learning Outcomes

Order Course Learning Outcomes
LO01 Learn the representations of Lie algebras.
LO02 learns the structure of Sl (n, C)Lie algebras and uses these algebras in applications.
LO03 Uses Cartan criterion
LO04 Learn the root space decomposition and root systems of Lie algebras.
LO05 Understands the existence of classical Lie algebras.
LO06 Categorize root systems


Relation with Program Learning Outcome

Order Type Program Learning Outcomes Level
PLO01 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in her area of expertise and other areas of mathematics. 5
PLO02 Bilgi - Kuramsal, Olgusal Knows in detail the relationship between the results in his area of ​​expertise and other areas of mathematics.
PLO03 Bilgi - Kuramsal, Olgusal Establishes new mathematical models with the help of the knowledge gained in the field of specialization.
PLO04 Bilgi - Kuramsal, Olgusal Has basic knowledge in all areas of mathematics. 3
PLO05 Bilgi - Kuramsal, Olgusal It presents the knowledge gained in different fields of mathematics and their relations with each other in the simplest and most understandable way.
PLO06 Bilgi - Kuramsal, Olgusal Effectively uses the technical equipment needed to express mathematics. 2
PLO07 Bilgi - Kuramsal, Olgusal poses original problems related to field and presents different solution techniques. 5
PLO08 Bilgi - Kuramsal, Olgusal carries out original and qualified scientific studies on the subject related to its field. 4
PLO09 Bilgi - Kuramsal, Olgusal Analyzes existing mathematical theories and develops new theories.
PLO10 Beceriler - Bilişsel, Uygulamalı Knows the teaching-learning techniques in areas of mathematics that require expertise and uses these techniques effectively at every stage of education.
PLO11 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği To have knowledge of a foreign language at a level to be able to follow foreign sources related to the field and to communicate verbally and in writing with foreign stakeholders. 3
PLO12 Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği presents and publishes its original works within the framework of scientific ethical rules for the benefit of its stakeholders. 2
PLO13 Yetkinlikler - Öğrenme Yetkinliği Adheres to the ethical rules required by its scientific title 5


Week Plan

Week Topic Preparation Methods
1 representations, modules and sub modules of Lie algebras Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
2 Inducible modules, irreducible modules, homomorphisms, Schur's Lemma Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
3 Representations of Sl (2, C) Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
4 Cartan criterion, Jordan decomposition, solubility test, Killing form Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
5 Semisimplicity test, derivatives of semisimple Lie algebras, Abstract Jordan decomposition Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
6 Root space decomposition, Cartan subalgebra, Definition of root space decomposition Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
7 Isomorphic subalgebras to SL (2, C), root chains and eigenvalues, cartan subalgebras and inner product spaces Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
8 Mid-Term Exam Study the relevant sections in the book Ölçme Yöntemleri:
Yazılı Sınav
9 Root systems, classification, base of the root systems, Cartan matrices and Dynkin diagrams Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
10 Classical Lie algebras, Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
11 so (2l, C), sp (2l, C) algebras Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
12 Classification of root systems Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
13 Simple Lie algebras, Serre's theorem Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
14 Irreducible representations of semi-linear Lie algebras, Universal enveloping algebras Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
15 Universal enveloping algebras Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
16 Algebras Study the relevant sections in the book Öğretim Yöntemleri:
Anlatım
17 Term Exams Study the relevant sections in the book Ölçme Yöntemleri:
Yazılı Sınav


Student Workload - ECTS

Works Number Time (Hour) Workload (Hour)
Course Related Works
Class Time (Exam weeks are excluded) 14 3 42
Out of Class Study (Preliminary Work, Practice) 14 5 70
Assesment Related Works
Homeworks, Projects, Others 0 0 0
Mid-term Exams (Written, Oral, etc.) 1 15 15
Final Exam 1 30 30
Total Workload (Hour) 157
Total Workload / 25 (h) 6,28
ECTS 6 ECTS