Second Level Degree in Mathematics
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REPRESENTATION THEORY OF GROUPS - 6 CFU
Teacher
Giovanna Carnovale
Scheduled Period
I Year - 2 Semester | 28/02/2022 - 11/06/2022
Hours: 48 (16 esercitazione, 32 lezione)
Prerequisites
Basic notions in linear algebra and group theory.
Target skills and knowledge
The student will learn the basic notions on complex representations of finite groups and the classification of complex semisimple Lie algebras.
Examination methods
Unless sanitary emergency situation forces to do otherwise, the exam will be written, based on a series of exercises.
Assessment criteria
Exams will be evaluated on basis of their completeness, correctness and exactness.
Course contents
Representations. Irreducible representations. Maschke's theorem. Orthogonality of characters. Induced representations. Frobenius reciprocity. Rappresentazioni Indotte, formual di Mackey. Reciprocita' di Frobenius. Frobenius-Scur Indicator. Compact groups. Linear algebraic groups and their Lie algebras. Solvable, nilpotent and semisimple Lie algebras. Cartan's criterion. Killing form. Weyl's theorem. Root space decomposition. Root systems. Classification of semisimple Lie algebras. Universal enveloping algebras. Finite dimensional irreducible representations of a semisimple Lie algebra.
Planned learning activities and teaching methods
Classroom lectures. Students will be provided with exercises that they are supposed to solve independently, alone or in groups, outside classroom time.
Additional notes about suggested reading
J.P. Serre, Répresentations Linéaires des Groupes Finis; (there exists also an English version);
J. Humphreys, Introduction to Lie algebras and Representation Theory, GTM 9 Springer
P. Etingof et al, Introduction to representation theory, AMS Macdonald's lectures in: Lectures on Lie groups and Lie algebras, Carter, Segal, Macdonald, Cambridge University Press, 1995