6 - Lie algebras of Lie groups, Kac-Moody groups, supergroups, and some specialized topics in finite- and infinite-dimensional Lie algebras

Abstract

In this chapter, first we discuss about some fundamental concepts in Lie groups, Lie algebras of Lie groups, Kac-Moody groups, supergroups, etc. and then some fundamental applications of Lie algebras and Lie groups to differential geometry, to number theory, and finally to differential equations. As an application to differential geometry is concerned, we define some differential operators and give the spectra of these differential operators on quadric hypersurface and quaternionic projective space. As an application to number theory, we explain about zeta and eta functions and give some results on spectral invariants of the zeta function of the Laplace-Beltrami operator acting on 1-forms on the above spaces and 2-forms on (4r − 1) dimensional sphere and eta function and spectral asymmetry of the operator B = ±(*d − d*) acting on 2-forms on (4r − 1) dimensional sphere. We also give some generalizations of Macdonald’s identities for some Kac-Moody algebras. As an application to differential equation, we discuss about Hirota bilinear differential operators, principal and homogeneous vertex operators and their corresponding Hirota bilinear equations and super Hirota bilinear equations for some simply and non-simply laced Kac-Moody algebras. Basic properties of Fermionic Fock space, Clifford algebra, Bosonic Fock space and Boson-Fermion correspondence along with some references to Quantum groups, String theory and Mathematical Physics are also given in this chapter.