Homework solutions for LSU MATH 7250, Representation Theory.
The course description is "This course is designed to give an introduction to representation theory, with an emphasis on Lie algebras and algebraic groups. The class is designed to be suitable both for students planning to specialize in representation theory and for those who need it for applications. It will start with an outline of the representation theory of finite groups over the complex numbers. We will then introduce complex algebraic groups and their Lie algebras. After discussing the basic theory of nilpotent, solvable, and semisimple Lie algebras, we will describe the classification of semisimple Lie algebras. We will continue with the universal enveloping algebra of a Lie algebra and the Poincaré-Birkhoff-Witt theorem. We will then cover highest-weight representations of Lie algebras, including Verma modules and finite-dimensional irreducible representations. We will also discuss the relationship between the representations of a semisimple algebraic group and the representations of its Lie algebra."
Problems mostly come from the instructor. Note that the problems only deal with (finite) group representations.
There may be errors or typos! You have been warned!