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 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
- <math>(c_1 A+c_2 B)[X]=c_1 A[X] + c_2 B[X]\,<math>
- <math>A[c_1 X + c_2 Y]=c_1 A[X] + c_2 A[Y]\,<math>
- <math>(-1)^{A[X]}=(-1)^A(-1)^X\,<math>
- <math>[A,B)[X]=A[B[X]]-(-1)^{AB}B[A[X]].\,<math>
Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above.