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Concept
Lie superalgebra
Summary
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading):
Super skew-symmetry:
The super Jacobi identity:
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.
One also sometimes adds the axioms for |x| = 0 (if 2 is invertible this follows automatically) and for |x| = 1 (if 3 is invertible this follows automatically). When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that the Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).
Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.
A graded Lie algebra (say, graded by Z or N) that is anticommutative and Jacobi in the graded sense also has a grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See note at graded Lie algebra for discussion.
Let be a Lie superalgebra. By inspecting the Jacobi identity, one sees that there are eight cases depending on whether arguments are even or odd. These fall into four classes, indexed by the number of odd elements:
No odd elements. The statement is just that is an ordinary Lie algebra.
One odd element. Then is a -module for the action .
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