Editing Lie superalgebra (section)

==Definition<!--'Lie superbracket' and 'Supercommutator' redirect here-->==
Formally, a Lie superalgebra is a nonassociative '''Z'''<sub>2</sub>-[[graded algebra]], or ''[[superalgebra]]'', over a [[commutative ring]] (typically '''R''' or '''C''') whose product [·,&nbsp;·], called the '''Lie superbracket'''<!--boldface per WP:R#PLA--> or '''supercommutator'''<!--boldface per WP:R#PLA-->, satisfies the two conditions (analogs of the usual [[Lie algebra]] axioms, with grading):

Super skew-symmetry:

:<math>[x,y]=-(-1)^{|x| |y|}[y,x].\ </math>

The super Jacobi identity:<ref>{{harvnb|Freund|1983|p=8}}</ref>

:<math>(-1)^{|x||z|}[x, [y, z]] + (-1)^{|y||x|}[y, [z, x]] + (-1)^{|z||y|}[z, [x, y]] = 0, </math>

where ''x'', ''y'', and ''z'' are pure in the '''Z'''<sub>2</sub>-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 <math>[x,x]=0</math> for |''x''|&nbsp;=&nbsp;0 (if 2 is invertible this follows automatically) and <math>[[x,x],x]=0</math> for |''x''|&nbsp;=&nbsp;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.