Graded Lie algebras in mathematics and physics (Bose--Fermi symmetry)

Graded Lie algebras have recently become a topic of interest in physics in the context of ''supersymmetries,'' relating particles of differing statistics. In mathematics, graded Lie algebras have been known for some time in the context of deformation theory. In this paper we discuss basic properties of graded Lie algebras and present various new constructs for producing examples of such algebras. In addition we present a short survey of the role played by graded Lie algebras in mathematics and review in some detail the recent applications of supersymmetry in the physics of particles and fields