Introduction to Lie Algebras

ABOUT THE COURSE:The Lie algebras are central to Lie theory. They are highly non-commutative and non-associative algebras. They were first introduced in 1870s by a Norwegian mathematician Marius Sophus Lie to study the concept of infinitesimal transformations. Their representations play an important role in theoretical Physics.The course will introduce finite dimensional Lie algebras. We will begin with the basic definitions and properties of finite dimensional Lie algebras and prove some fundamental results about nilpotent and solvable Lie algebras. Then we will prove the Cartan’s criteria for solvability and semi simplicity. Finally, we will move on to the structure theory of semi-simple Lie algebras and prove their root space decomposition.INTENDED AUDIENCE: NonePREREQUISITES: First course in Linear algebraINDUSTRY SUPPORT: Nil

Week 1: Basis definitions, examples and some elementary propertiesWeek 2:Ideals, Homomorphisms and Quotient algebrasWeek 3:Low dimensional Lie algebras: classifications up to dimension 3Week 4:Abelian, Nilpotent, Solvable Lie algebrasWeek 5:Subalgebras of general linear Lie algebra and the invariance lemmaWeek 6:Representations of nilpotent Lie algebras: Engel’s theoremWeek 7:Representations of solvable Lie algebras: Lie’s theorem Week 8:General representation theory: irreducible/indecomposable representations, Schur lemmaWeek 9:Classification of irreducible representations of sl_2Week 10:Cartan’s criteria for solvability and semi-simplicityWeek 11:Jordan decomposition and abstract Jordan decompositionWeek 12:Cartan subalgebras and root space decomposition of semi-simple Lie algebras