Representation of a Lie superalgebra - Definition and Overview

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.

See also representation of a Lie algebra, representation of a Hopf algebra, Lie superalgebra, group representation, graded vector space