representation of a lie superalgebra

In the theory of Lie superalgebras, a representation of a Lie superalgebra L is the action of L upon a Z₂-graded vector space V such that if A and B are any two pure elements of L (remember that L is Z₂-graded) 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 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. See also representation of a Lie algebra, representation of a Hopf algebra, Lie superalgebra, group representation, graded vector space

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