PURE MTH 7002  Pure Mathematics Topic B
North Terrace Campus  Semester 1  2021

General Course Information
Course Details
Course Code PURE MTH 7002 Course Pure Mathematics Topic B Coordinating Unit Mathematical Sciences Term Semester 1 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Y Assessment Ongoing assessment, exam Course Staff
Course Coordinator: Dr David Baraglia
Course Timetable
The full timetable of all activities for this course can be accessed from Course Planner.

Learning Outcomes
Course Learning Outcomes
In 2021, the topic of this course is Lie Algebras.
Outline
Lie algebras and Lie groups are fundamental concepts which arise in many areas of mathematics and theoretical physics. Lie groups arise as symmetry groups of continuous structures, just as finite groups arise as symmetries of discrete structures. The group of rotations of the unit sphere in R^3 is an example of a Lie group. Lie algebras and Lie groups arise in algebra, geometry, topology, differential equations and number theory. In physics Lie algebras and Lie groups are important in describing symmetries of physical systems and in gauge theories.
An important aspect of the theory is that to each Lie group there is a corresponding Lie algebra, which is to be thought of as a kind of
infinitesimal linearisation of the Lie group. Due to their linear structure, Lie algebras are considerably easier to work with than Lie groups as they can be studied using tools from linear algebra. Furthermore, the structure of the Lie algebra almost completely determines the structure of the corresponding Lie group. In this way the study of Lie groups can largely be reduced to the study of Lie algebras.
The main goal of this course will be to study Lie algebras (and therefore, indirectly, Lie groups), understand their basic structure theory, and to obtain the classification of complex semisimple Lie algebras via root systems and Dynkin diagrams. The main emphasis in this course will be on the structure theory of Lie algebras, however we will also examine the relation between Lie algebras and Lie
groups.
Topics
1. Basic concepts of Lie algebras and Lie groups.
2. The relation between Lie algebras and Lie groups.
3. Solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem.
4. The Killing form, semisimple Lie algebras, Cartan's criterion.
5. Cartan subalgebras, root space decompositions.
6. Root systems and their classification.
7. The classification of complex semisimple Lie algebras.
Learning Outcomes
On successful completion of this course, students will be able to:
1. Understand the definition of Lie algebras and related concepts.
2. Understand how Lie algebras arise from Lie groups.
3. Distinguish between nilpotent, solvable, semisimple and simple Lie algebras using effectively several criteria such as Lie's theorem,
Engel's theorem, Cartan's criterion.
4. Recognise Cartan subalgebras and use them to obtain root space decompositions of complex semisimple Lie algebras.
5. Use root systems and Dynkin diagrams in order to classify the complex semisimple Lie algebras.
Prerequisites
The main prerequisite for this course is a good understanding of linear algebra at the level of Algebra II. Basic knowledge of group theory will also be helpful. Pure Math Topic A (Differential Geometry) will be useful in understanding the relationship between Lie groups and Lie algebras, but not mandatory.University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
 informed and infused by cutting edge research, scaffolded throughout their program of studies
 acquired from personal interaction with research active educators, from year 1
 accredited or validated against national or international standards (for relevant programs)
all Critical thinking and problem solving
 steeped in research methods and rigor
 based on empirical evidence and the scientific approach to knowledge development
 demonstrated through appropriate and relevant assessment
all 
Learning Resources
Required Resources
This is a reading course. The course will make use of the following textbook, which is available online through the university library.
K. Erdmann and M. J. Wildon, Introduction to Lie algebras.
Supplementary notes on Lie groups and their relation to Lie algebras will also be provided.Recommended Resources
There are many excellent references for Lie groups and Lie algebras. The most suitable ones for this course are:
K. Erdmann and M. J. Wildon, Introduction to Lie algebras (this will be our main reference for Lie algebras)
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (a slightly more advanced reference for Lie algebras)
H. Samelson, Notes on Lie Algebras (covers much the same material as Humphreys)
A. K. Knapp, Lie Groups, Beyond an Introduction (contains much more material than we will cover in this course, but is very well written)
W. Fulton, J. Harris, Representation Theory, A First Course (covers somewhat different material, but is heavily example based so may be
useful as a learning resource for Lie algebras and Lie groups)
Online Learning
The course will have an active MyUni website. 
Learning & Teaching Activities
Learning & Teaching Modes
This is a reading course. Students are expected to read and engage with the assigned reading material. There will be weekly face to face
meetings in which students can discuss the material with the course coordinator. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.Workload
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Study 30 90 Assignments 5 66 Total 156 Learning Activities Summary
1. Basic concepts of Lie algebras and Lie groups.
2. The relation between Lie algebras and Lie groups.
3. Solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem.
4. The Killing form, semisimple Lie algebras, Cartan's criterion.
5. Cartan subalgebras, root space decompositions.
6. Root systems and their classification.
7. The classification of complex semisimple Lie algebras. 
Assessment
The University's policy on Assessment for Coursework Programs is based on the following four principles:
 Assessment must encourage and reinforce learning.
 Assessment must enable robust and fair judgements about student performance.
 Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
 Assessment must maintain academic standards.
Assessment Summary
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% All Homework assignments Formative and summative One week after assigned 30% All Assessment Related Requirements
An aggregate score of 50% is required to pass the course.Assessment Detail
There will be five homework assignments distributed approximately once every two weeks across the semester. There will also be a final
examination.Submission
Homework assignments must be submitted on MyUni as pdf files. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.Course Grading
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 149 Fail P 5064 Pass C 6574 Credit D 7584 Distinction HD 85100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending Further details of the grades/results can be obtained from Examinations.
Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.
Final results for this course will be made available through Access Adelaide.

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