- Representation of a Lie superalgebra
In the mathematical field of

representation theory , a**representation of a Lie superalgebra**is an action ofLie superalgebra "L" on a**Z**_{2}-graded vector space "V", such that if "A" and "B" are any twopure element s of "L" and "X" and "Y" are any two pure elements of "V", then:$(c\_1\; A+c\_2\; B)\; [X]\; =c\_1\; A\; [X]\; +\; c\_2\; B\; [X]\; ,$

:$A\; [c\_1\; X\; +\; c\_2\; Y]\; =c\_1\; A\; [X]\; +\; c\_2\; A\; [Y]\; ,$

:$(-1)^\{A\; [X]\; \}=(-1)^A(-1)^X,$

:$[A,B)\; [X]\; =A\; [B\; [X]\; -(-1)^\{AB\}B\; [A\; [X]\; .,$

Equivalently, a representation of "L" is a

**Z**_{2}-graded representation of theuniversal 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 aninvolutive antilinear map ^{*}such that * respects the grading and: [a,b]

^{*}= [b^{*},a^{*}]A

unitary representation of such a Lie algebra is a**Z**_{2}gradedHilbert space which is a representation of a Lie superalgebra as above together with the requirement thatself-adjoint elements of the Lie superalgebra are represented byHermitian transformations.This is a major concept in the study of

supersymmetry together withrepresentation 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)^{La}L^{*}[a^{*}] ) and**H**is the unitary rep and also,**H**is aunitary 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,:L [a|ψ>)] =(L [a] )|ψ>+(-1)

^{La}a(L [|ψ>] )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:L [a] =La-(-1)

^{La}aLThis approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary

Grassmann number s.**ee also***

Graded vector space

*Lie algebra representation

*Representation theory of Hopf algebras

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