Unit I

Basic Concepts Definition and Examples, Lie Algebra of Derivations, Adjoint Representation,
Structure Constants, Direct Sums, Homomorphism and Isomorphisms, Ideals, Centre and Derived Algebra of a Lie Algebra, Simple Lie Algebras, The Normalizer of a Subalgebra and Centralizer of a Subset in Lie Algebras, Automorphism and Inner Automorphism of a Lie Algebra.( Book 1, Chapters 1 and 2)

Unit II

Descending Central Series of a Lie Algebra, Nilpotent Lie Algebras. Derived Series of a Lie Algebra, Radical of a Lie Algebra, Solvable Lie Algebras, Engel’s Theorem.
(Book 1, Chapter 3)

Unit III

Semisimple Lie Algebras Theorems of Lie and Cartan, Jordan- Chevalley Decomposition,Cartan’s Criterion. ( Book 1, Chapter 4)

Unit IV

Killing Form, Inner Derivations, Abstarct Jordan Decomposition, Complete Reducibility of Lie algebras.. ( Book 1, Chapter 5)

Unit V

The Weyl Group, Root Systems. ( Book 1, Chapter 10)

Course Outcome:

CO 1: To understand the concept of Lie algebra and to know the substructures and operations on them.
CO 2: To familiarize nilpotent and solvable Lie algebras and prove the Engel’s theorem
CO 3: To understand theorems on Semi simple Lie algebras and their applications .
CO 4: To derive various decomposition theorems on Lie algebras
Co 5: To understand the classification of Lie algebras through Dynkin diagrams.

1. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1980.
2. Jacobson, Lie Algebras, Dover, 1979.
3. J.P. Serre, Lie Algebras and Lie Groups, Benjamin, 1965 (Translated from French).