Representation of a Lie superalgebra

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
Equivalently, a representation of "L" is a Z2-graded representation of the universal enveloping algebra of "L" which respects the third equation above.
Unitary representation of a star Lie superalgebra.
A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and 
A unitary representation of such a Lie algebra is a Z2 graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint elements of the Lie superalgebra are represented by Hermitian transformations.
This is a major concept in the study of supersymmetry together with representation of a Lie superalgebra on an algebra. Say A is an *-algebra representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L[a]*=-(-1)LaL*[a*]) and H is the unitary rep and also, H is a unitary representation of A.
These three reps are all compatible if for pure elements a in A, |ψ> in H and L in the Lie superalgebra,
Sometimes, the Lie superalgebra is embedded within A in the sense that there is a homomorphism from the universal enveloping algebra of the Lie superalgebra to A. In that case, the equation above reduces to
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers.