Representation of a Lie superalgebra

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In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z₂-graded vector space V, such that if A and B are any two pure elements of L and X and Y are any two pure elements of V, then

${\displaystyle (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X}$

${\displaystyle A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y}$

${\displaystyle (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}}$

${\displaystyle [A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).}$

Equivalently, a representation of L is a Z₂-graded representation of the universal enveloping algebra of L which respects the third equation above.