Course Identification

A course in Linear Algebra (including Jordan forms).

• The course will work  using the following scheme (until the frontal teaching will be allowed):

   

  the lecture notes and the homework assignments will be sent each week;

   

  a virtual questions and answers session will be held during 

  lecture hours  Monday 9:15-11:00; the first session will be on 20/04/2020;

   

  the homework solutions should be sent by email to Maria or left in her mailbox.

   

  Please, send an email to Maria if you would like to receive the lecture notes and

  the homework assignments. Please, indicate if you wish to participate in Q& A sessions.

   

  In order to pass the course one has to submit all homeworks with at least

  half of the problems solved  and take a final exam. The participants will have an opportunity to

  discuss the homework assignment on the virtual Q & A session which will be held 

  at least 3 days before the deadline of the assignment.

Syllabus: 

• Definition of a Lie algebra, examples.
• Representations, representation theory of sl(2)
• Solvable and nilpotent Lie algebras, Engel's Theorem, Lie's Theorem
• Simplicity and semi-simplicity, Killing form, Cartan's criterion
• Complete reducibility of representations, Casimir element
• Root systems, Weyl group
• Cartan matrix, Dynkin diagrams
• Classification of complex semi-simple Lie algebras
• Universal enveloping algebra, Poincare-Birkoff-Witt Theorem
• Verma modules
• Weyl Character formula

Upon successful completion of this course students should be able to
demonstrate an understanding of the concepts of structure theory of complex semisimple Lie algebras and their representations.

Suggested Reading: J. E. Humphreys "Introduction to Lie Algebras and Representation Theory"