Editing Representation of a Lie superalgebra

{{unreferenced|date=May 2014}}
In the [[mathematics|mathematical]] field of [[representation theory]], a '''representation of a Lie superalgebra''' is an [[semigroup action|action]] of [[Lie superalgebra]] ''L'' on a [[graded vector space|'''Z'''<sub>2</sub>-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

:<math>(c_1 A+c_2 B)\cdot X=c_1 A\cdot X + c_2 B\cdot X</math>

:<math>A\cdot (c_1 X + c_2 Y)=c_1 A\cdot X + c_2 A\cdot Y</math>

:<math>(-1)^{A\cdot X}=(-1)^A(-1)^X</math>

:<math>[A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).</math>

Equivalently, a representation of ''L'' is a '''Z'''<sub>2</sub>-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 [[Involution (mathematics)|involutive]] [[antilinear]] [[map]] * such that * respects the grading and 

:[a,b]*=[b*,a*].

A [[unitary representation]] of such a Lie algebra is a '''Z'''<sub>2</sub> [[graded vector space|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 [[star-algebra|*-algebra]] representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L[a]*=-(-1)<sup>La</sup>L*[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,

:L[a|ψ>)]=(L[a])|ψ>+(-1)<sup>La</sup>a(L[|ψ>]).

Sometimes, the Lie superalgebra is [[embedding|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)<sup>La</sup>aL.

This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary [[Grassmann number]]s.

==See also==
* [[Graded vector space]]
* [[Lie algebra representation]]
* [[Representation theory of Hopf algebras]]

[[Category:Representation theory of Lie algebras]]
[[Category:Supersymmetry]]



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