4 - Lie superalgebras

Abstract

In Mathematics, graded Lie algebras were known in the context of deformation theory. Graded Lie algebras also appear in Physics in the context of “Supersymmetrices” relating to particles of differing statistics. The first basic example of graded Lie algebras was provided by Nijenhuis [120] and then by Frolicher and Nijenhuis [121]. In a paper of “Berezin and Kats [122],” Lie superalgebras appear as Lie algebras of certain generalized groups, called Lie supergroups. A satisfactory theory similar to Lie’s theory, has been developed on the connection between Lie supergroups and Lie superalgebras [123, 124]. Kac [34] gave a construction of the theory of “Lie superalgebras” (as the Physicists call them, “Z2-graded Lie algebras”) on the lines of Lie algebras. A detailed exposition of the theory of Lie superalgebras can also be seen in Scheunert [35]. Lie superalgebras also appear in Cohomology theories. Gerstenhaber [125] described cohomology structure of an associative ring.