Representation of a Lie superalgebra

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In mathematics, particularly in the theory of Lie superalgebras, a representation of a Lie superalgebra L is the action of 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

(c_1 A+c_2 B)[X]=c_1 A[X] + c_2 B[X]\,
A[c_1 X + c_2 Y]=c_1 A[X] + c_2 A[Y]\,
(-1)^{A[X]}=(-1)^A(-1)^X\,
[A,B)[X]=A[B[X]]-(-1)^{AB}B[A[X]].\,

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

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