You are in:: Start > M3036
Site map
Groups and Lie algebras
| Code: | M3036 | Acronym: | M3036 |
| Keywords | |
|---|---|
| Classification | Keyword |
| OFICIAL | Mathematics |
Instance: 2021/2022 - 2S 
| Active? | Yes |
| Responsible unit: | Department of Mathematics |
| Course/CS Responsible: | Bachelor in Mathematics |
Cycles of Study/Courses
| Acronym | No. of Students | Study Plan | Curricular Years | Credits UCN | Credits ECTS | Contact hours | Total Time |
|---|---|---|---|---|---|---|---|
| L:M | 16 | Official Study Plan | 2 | - | 6 | 56 | 162 |
| 3 |
Teaching language
Suitable for English-speaking studentsObjectives
Introduction of the first basic notions related to matrix Lie groups and their Lie algebras.Learning outcomes and competences
Upon completing this course, the student should:
- master concepts, methods and basic results of matrix Lie groups and Lie algebras;
- be able to analyze and solve problems within the scope of matrix Lie groups and Lie algebras, using the methods and results that best apply to the problem under study;
- have adequate preparation to pursue studies in areas of Mathematics that require theory of matrix Lie groups and Lie algebras;
- be able to efficiently communicate their problem solutions and understanding of the subject.
- master concepts, methods and basic results of matrix Lie groups and Lie algebras;
- be able to analyze and solve problems within the scope of matrix Lie groups and Lie algebras, using the methods and results that best apply to the problem under study;
- have adequate preparation to pursue studies in areas of Mathematics that require theory of matrix Lie groups and Lie algebras;
- be able to efficiently communicate their problem solutions and understanding of the subject.
- master concepts, methods and basic results of group theory and matrix Lie algebras;
- be able to analyze and solve problems within the scope of group theory and Lie algebras of matrices, using the methods and results that best apply to the problem under study;
- have adequate preparation to pursue studies and research in areas of Mathematics that integrate or use the theory of groups and Lie matrix algebras;.
- be able to efficiently communicate their problem solutions and understanding of the subject matter." data-crosslingual-hint="" data-location="2" data-enable-toggle-playback-speed="true">
Working method
PresencialPre-requirements (prior knowledge) and co-requirements (common knowledge)
Linear Algebra (I e II), Calculus of several variables (Analysis III; Topological notions in R^n), Abstract Algebra; Basic Complex Analysis.Program
I. Matrix Lie Groups: Definitions; Examples (general and special linear groups, unitary and orthogonal groups, generalized orthogonal groups, Lorentz groups, symplectic groups, Euclidean group, Heisenberg group); Compactness and Connectedness; Tori; Maximal Tori of Compact Matrix Lie Groups; Conjugates of a Maximal Torus; SU(2) and SO(3).
II. Exponential: Exponential of a Matrix; Properties; Computing the Exponential; Matrix Logarithm; Properties; Further Properties of the Exponential; Polar Decomposition.
III. Lie Algebras: Definitions and First Examples; Simple, Solvable, and Nilpotent Lie Algebras; Lie Algebra of a Matrix Lie Group; Examples; Lie Group and Lie Algebra Homomorphisms; Adjoint Map; Complexification of a Real Lie Algebra; Exponential Map; Exponential Map as a Local Chart; Lie Algebras of Matrix Lie Groups and Tangent Spaces ast the Identity; Lie Algebras of Maximal Tori and Aplications.
IV. Basic Representation Theory: Representations; Properties; Examples; Direct Sums and Tensor Products of Representations; Dual Representations; Complete Reducibility; Schur's Lemma; Representations of sl(2,C).
II. Exponential: Exponential of a Matrix; Properties; Computing the Exponential; Matrix Logarithm; Properties; Further Properties of the Exponential; Polar Decomposition.
III. Lie Algebras: Definitions and First Examples; Simple, Solvable, and Nilpotent Lie Algebras; Lie Algebra of a Matrix Lie Group; Examples; Lie Group and Lie Algebra Homomorphisms; Adjoint Map; Complexification of a Real Lie Algebra; Exponential Map; Exponential Map as a Local Chart; Lie Algebras of Matrix Lie Groups and Tangent Spaces ast the Identity; Lie Algebras of Maximal Tori and Aplications.
IV. Basic Representation Theory: Representations; Properties; Examples; Direct Sums and Tensor Products of Representations; Dual Representations; Complete Reducibility; Schur's Lemma; Representations of sl(2,C).
Mandatory literature
Brian C. Hall; Lie groups, Lie algebras, and representations. ISBN: 0-387-40122-9Complementary Bibliography
Kristopher Tapp; Matrix groups for undergraduates. ISBN: 9780821837856José Carlos Santos; Grupos e álgebras de Lie. ISBN: 978-989-8481-04-7
Anthony W. Knapp; Lie groups beyond an introduction. ISBN: 0-8176-3926-8
Roger Godement; Introduction a la theorie des groupes de lie. tome i
Roger Godement; Introduction a la theorie des groupes de lie. tome ii
Teaching methods and learning activities
Contact hours consist of theoretical and practical classes, allowing the teacher to organize and manage the available time for the presentation of hte contents, as well as presenting examples and solving exercises. The active participation of students is encouraged.Evaluation Type
Distributed evaluation with final examAssessment Components
| designation | Weight (%) |
|---|---|
| Exame | 80,00 |
| Trabalho escrito | 20,00 |
| Total: | 100,00 |
Amount of time allocated to each course unit
| designation | Time (hours) |
|---|---|
| Estudo autónomo | 106,00 |
| Frequência das aulas | 56,00 |
| Total: | 162,00 |
Eligibility for exams
Attendance is not mandatory.Calculation formula of final grade
- Two written homeworks with a weight of 10% each, making a total weight of 20% of the final grade.- Final exam with a weight of 80% of the final grade.
Any final grade higher than 17 points may also be subject to an extra assessment test.