Studies in mathematical aspects of super Lie algebras

Abstract

One of the most novel concepts in elementary particle physics is that of supersymmetry which ascribes a possible fundamental symmetry between fermions and bosons obeying different statistics. It was an assumption till then that it is impossible to form multiplets of particles with different spins; With the advent of notion of super symmetry, it is now possible to have super multiplets which accomodate both fermions and bosons. Because of the different spin and statistics of the constituents of the super multiplet, it is obvious that there is a need for a new algebraic structure which relates the states of bosons and fermions. This is precisely provided by Super Lie algebras and Z 2 graded Lie algebras. The frame work of Super Lie Algebras and Gauge principles seems to provide us with the necessary tools for realizing an unification of all the interactions - strong, weak, electromagnetic and Gravitation. The author pays an immense interest in mathematics of the graded algebras particularly, about their structure and their representations. This thesis concentrates on the study of the structure of Super Lie Algebras which was initiated by Corwin et al., Kaplansky, Djokovic, and Hochschild and Scheunert. Weight root theorem is discussed in this thesis, to relate the weights of the representation to the roots of the Lie Algebra. The C-Theorem to relate its coefficients also is discussed, and it is shown that C-Theorem contains an error; A modified C-Theorem is then suggested. Pais and Rittenberg of SSGLA concluded in their Classification that there is only one semisimple graded Lie Algebra, with even part being the Symplectic Lie Algebra. It is shown that inspite of the error in C-Theorem, their classification is not affected.