Editing Lie superalgebra (section)

== Properties ==
Let <math>\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1</math> 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:<ref>{{harvnb|Varadarajan|2004|p=89}}</ref>

# No odd elements. The statement is just that <math>\mathfrak g_0</math> is an ordinary Lie algebra.
# One odd element. Then <math>\mathfrak g_1</math> is a <math>\mathfrak g_0</math>-module for the action <math>\mathrm{ad}_a: b \rightarrow [a, b], \quad a \in \mathfrak g_0, \quad b, [a, b] \in \mathfrak g_1</math>.
# Two odd elements. The Jacobi identity says that the bracket <math>\mathfrak g_1 \otimes \mathfrak g_1 \rightarrow \mathfrak g_0</math> is a ''symmetric'' <math>\mathfrak g_1</math>-map.
# Three odd elements. For all <math>b \in \mathfrak g_1</math>, <math>[b,[b,b]] = 0</math>.

Thus the even subalgebra <math>\mathfrak g_0</math> of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while <math>\mathfrak g_1</math> is a [[representation of a Lie algebra|linear representation]] of <math>\mathfrak g_0</math>, and there exists a [[symmetric]] <math>\mathfrak g_0</math>-[[equivariant]] [[linear map]] <math>\{\cdot,\cdot\}:\mathfrak g_1\otimes \mathfrak g_1\rightarrow \mathfrak g_0</math> such that,

:<math>[\left\{x, y\right\},z]+[\left\{y, z\right\},x]+[\left\{z, x\right\},y]=0, \quad x,y, z \in \mathfrak g_1.</math>

Conditions (1)&ndash;(3) are linear and can all be understood in terms of ordinary Lie algebras. Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra (<math>\mathfrak g_0</math>) and a representation (<math>\mathfrak g_1</math>).