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Information
| Code | MT016 |
| Name | Lie Algebras |
| Term | 2024-2025 Academic Year |
| Term | Spring |
| 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 Instructor |
1 |
Course Goal / Objective
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 | ||