Graded Lie Algebras
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Graded Lie Algebras (GLAs) were first introduced in Mathematics in 1961 by Nijenhuis and Richardson as a generalization of the notion of a Lie algebra which proved to be of use in the theory of deformation of structures. However, it was not until the advent of the applications in physics, in the guise of supersymmetry and supergravity, that research into their properties really got a firm start. What we intend to do in this chapter is to summarize briefly some of the properties of GLAs which are of importance from both the physicist’s and the mathematician’s point of view. Basically, as we know, a Lie algebra consists of objects which have a well defined ‘Lie algebra bracket’ defined on them. The basic properties of such brackets are well known — they are antisymmetric, bilinear, and satisfy Jacobi identities. Such Lie algebras are suitable for describing quantum field theoretic operators which are bosonic in nature, i.e. which satisfy commutation relations among themselves. However, fermionic field theoretic operators satisfy anti-commutation rules among themselves. Hence, we have a need for a structure which admits both kinds of operators, bosonic and fermionic, and both kinds of ‘brackets’, commutators and anticommutators. Such a structure is precisely the one provided by GLAs A GLA therefore, should consist of two kinds of objects: bosonic or even generators and fermionic or odd generators which have the following relations between them.
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- The mathematical theory of GLAs is covered in detail by M. Scheunert, The Theory of Lie Superalgebras, Springer Lecture Notes in Maths 716 (1979).Google Scholar