## PURE MTH 4066 - Pure Mathematics Topic E - Honours

### North Terrace Campus - Semester 2 - 2022

This course is available for students taking an honours degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.

• General Course Information
##### Course Details
Course Code PURE MTH 4066 Pure Mathematics Topic E - Honours School of Mathematical Sciences Semester 2 Undergraduate North Terrace Campus 3 Y Honours students only This course is available for students taking an honours degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.
##### 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 2022, 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. For example, the group of rotations of the unit sphere in R^3 is 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 essential.
##### 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)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

1, 2

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

1, 2, 3

Attribute 3: Teamwork and communication skills

Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

3

Attribute 8: Self-awareness and emotional intelligence

Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

3
• 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
Course information and resources will be posted on MyUni.

• Learning & Teaching Activities
##### Learning & Teaching Modes
Students are expected to read and engage with the assigned reading material. There will be a weekly workshop with a mix of lecturing, students working on problems, together and with guidance from the lecturer, and consulting. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload Hours Workshops 12 24 Assignments 5 50 Self-study 82 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:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 60% all Homework assignments Formative and summative Weeks 3,5,7,9,11 40% all
##### Assessment Related Requirements
A mark of 50 is required to pass the course.
##### Assessment Detail
There will be five homework assignments, due in Weeks 3, 5, 7, 9 and 11.

##### Submission
Homework assignments should be submitted via MyUni.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
GradeGrade reflects following criteria for allocation of gradeReported on Official Transcript
Fail A mark between 1-49 F
Third Class A mark between 50-59 3
Second Class Div B A mark between 60-69 2B
Second Class Div A A mark between 70-79 2A
First Class A mark between 80-100 1
Result Pending An interim result RP
Continuing Continuing CN

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.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

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