Editing Lie superalgebra (section)

==Examples==
Given any [[associative superalgebra]] <math>A</math> one can define the supercommutator on homogeneous elements by

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

and then extending by linearity to all elements. The algebra <math>A</math> together with the supercommutator then becomes a Lie superalgebra. The simplest example of this procedure is perhaps when <math>A</math> is the space of all linear functions <math>\mathbf {End}(V)</math> of a super vector space <math>V</math> to itself. When <math>V = \mathbb K^{p|q}</math>, this space is denoted by <math>M^{p|q}</math> or <math>M(p|q)</math>.<ref>{{harvnb|Varadarajan|2004|p=87}}</ref> With the Lie bracket per above, the space is denoted <math>\mathfrak {gl}(p|q)</math>.<ref>{{harvnb|Varadarajan|2004|p=90}}</ref>

A [[Poisson algebra]] is an associative algebra together with a Lie bracket. If the algebra is given a '''Z'''<sub>2</sub>-grading, such that the Lie bracket becomes a Lie superbracket, then one obtains the [[Poisson superalgebra]]. If, in addition, the associative product is made [[supercommutative]], one obtains a supercommutative Poisson superalgebra.

The [[Whitehead product]] on homotopy groups gives many examples of Lie superalgebras over the integers.

The [[super-Poincaré algebra]] generates the isometries of flat [[superspace]].