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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>
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:<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 <sup>*</sup> [[Lie superalgebra]] is a complex Lie superalgebra equipped with an [[Involution (mathematics)|involutive]] [[antilinear]] [[map]] <sup>*</sup> such that * respects the grading and  
 
:[a,b]<sup>*</sup>=[b<sup>*</sup>,a<sup>*</sup>].
 
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]<sup>*</sup>=-(-1)<sup>La</sup>L<sup>*</sup>[a<sup>*</sup>]) 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]]
 
 
 
{{algebra-stub}}

Latest revision as of 09:41, 9 December 2014


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