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.
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.
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.
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