Course Identification

A course in Linear Algebra (including  the Jordan forms). 

• The course will work  using the following scheme:

• the lecture notes and the homework assignments will be sent each week:

• 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. 

   

  In order to pass the course one has to submit all homework assignments  with at least half of the problems solved  and take a final exam. The participants will have an opportunity to discuss the homework assignment during the practical sessions  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
• 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"