Lie Groups
A.Y. 2026/2027
Learning objectives
The course aims at providing the basic notions of Lie Groups and Lie Algebras.
Expected learning outcomes
The expected learning outcomes regard the knowledge and the ability to use Lie groups and their fundamental topological and differential properties.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
LIE GROUPS
· Vector Fields and Associated Flows; Involutive and Integrable Distributions and the Frobenius Theorem;
· Definition of Lie Groups and Initial Examples;
· Considerations on Coverings of Lie Groups;
· Fundamental Group of Lie Groups;
· Lie Algebras, Right- and Left-Invariant Fields; · Theorems on Lie Groups and Algebras: Correspondences; · Representation Ad and ad, Killing Form;
· Considerations on Abelian Lie Groups.
ACTIONS OF LIE GROUPS
· Free and Proper Actions;
· Actions of Compact Groups. Unimodular Groups. Existence of Haar Measures;
· The Slice Theorem and Ideas from the Proof;
· Classification of Orbits;
· Symplectic Manifolds, Recall of Principal and Associated Bundles;
· Hamiltonian Actions, Momentum Map;
· Symplectic reductions, Marsden-Weintein theorem;
· Toric Symplectic Manifolds, Delzant's theorem.
· Vector Fields and Associated Flows; Involutive and Integrable Distributions and the Frobenius Theorem;
· Definition of Lie Groups and Initial Examples;
· Considerations on Coverings of Lie Groups;
· Fundamental Group of Lie Groups;
· Lie Algebras, Right- and Left-Invariant Fields; · Theorems on Lie Groups and Algebras: Correspondences; · Representation Ad and ad, Killing Form;
· Considerations on Abelian Lie Groups.
ACTIONS OF LIE GROUPS
· Free and Proper Actions;
· Actions of Compact Groups. Unimodular Groups. Existence of Haar Measures;
· The Slice Theorem and Ideas from the Proof;
· Classification of Orbits;
· Symplectic Manifolds, Recall of Principal and Associated Bundles;
· Hamiltonian Actions, Momentum Map;
· Symplectic reductions, Marsden-Weintein theorem;
· Toric Symplectic Manifolds, Delzant's theorem.
Prerequisites for admission
Geometry 1,2,3,4. Some results of the theory of Covering spaces (I will recall them)
Teaching methods
Frontal lessons
Teaching Resources
Lie Groups and geometric aspects of isometric actions di R. Bettiol e M. Alexandrino (Springer, Cham, 2015. x+213 pp.)
Notes of Podestà and Spiro (on Ariel) (https://agorigl.ariel.ctu.unimi.it/v5/home/Default.aspx)
Notes of Abbena Console Garbiero (on Ariel)https://agorigl.ariel.ctu.unimi.it/v5/home/Default.aspx
Notes of Podestà and Spiro (on Ariel) (https://agorigl.ariel.ctu.unimi.it/v5/home/Default.aspx)
Notes of Abbena Console Garbiero (on Ariel)https://agorigl.ariel.ctu.unimi.it/v5/home/Default.aspx
Assessment methods and Criteria
Learning assessment is achieved through an oral exam covering the entire program of the course. Students may optionally prepare a seminar on topics suggested by the instructor that are related to the course but not covered in the course. In this case, during the oral exam, students will begin with the seminar presentation and then be questioned on other topics covered during the course.
The final grade is an overall assessment of the presentation and the rest of the exam.
The final grade is an overall assessment of the presentation and the rest of the exam.