An approach to dynkin diagrams associated with KAC-moody superalgebra

Abstract

In the present project report, a sincere report has been made to construct and study the basic portions related to Simple Lie Algebras, Lie superalgebras and Kac-Moody (super-) algebras and their corresponding Dynkin Diagrams.

In chapter-1, I have given the precise definitions of Lie Algebra and some of the terms related to Lie algebra, i.e. subalgebras, ideals, commutativity, solvability, nilpotency etc. Also, I have done the classifications of Classical Lie algebras.
In chapter-2, I made a review over the basics of Representation Theory, i.e. structure constants, modules, reflections in a Euclidean space, root systems (simple roots) and their corresponding root diagrams. Then I have discussed the formation of Dynkin Diagrams associated with the roots of the simple lie algebras.
After that, in chapter-3, the introductory parts of Lie superalgebras, i.e. Z2 graded algebra, definition of lie superalgebra, modules, the killing form, root systems and simple root systems (distinguished root systems) are addressed. In this section, also the classifications of simple lie superalgebras are plotted and the Dynkin Diagrams related to the Basic Lie superalgebras are presented.
In chapter-4, I gave the necessary theory based on Kac-Moody Lie superalgebras and their classifications. Then the definition of the extended Dynkin diagrams for Affinization of Kac-Moody superalgebras and the Dynkin Diagrams associated with the affine Kac-Moody superalgebras are provided.