Super Jacobi Identity

In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z₂ grading. 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).

Definition

Formally, a Lie superalgebra is a nonassociative Z₂-graded algebra, or ''superalgebra'', over a commutative ring (typically R or C) whose product �, Â· called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading):

Super skew-symmetry:

: ,y-(-1)^ ,x\

The super Jacobi identity:

:(-1)^ , [y, z + (-1)^[y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x + (-1)^[z, [x, y">,_z">,_[y,_z<_a>_+_(-1)^[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x + (-1)^[z, [x, y = 0,

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 ,yis the sum of degree of x and y modulo 2.

One also sometimes adds the axioms ''x''"> = 0 (if 2 is invertible this follows automatically) and $x,x$x0 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 $Z_2$ 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.

Properties

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ 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 $\mathfrak g_0$ is an ordinary Lie algebra.
# One odd element. Then $\mathfrak g_1$ is a $\mathfrak g_0$-module for the action \mathrm_a: b \rightarrow , b \quad a \in \mathfrak g_0, \quad b, , b\in \mathfrak g_1.
# Two odd elements. The Jacobi identity says that the bracket $\mathfrak g_1 \otimes \mathfrak g_1 \rightarrow \mathfrak g_0$ is a ''symmetric'' $\mathfrak g_1$-map.
# Three odd elements. For all $b \in \mathfrak g_1$, ,[b,b = 0.

Thus the even subalgebra $\mathfrak g_0$ of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while $\mathfrak g_1$ is a representation of a Lie algebra">linear representation