Unitary representations of a star Lie superalgebra
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In mathematics, a * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and
- [a,b]*=[b*,a*]
A unitary representation of such a Lie algebra is a Z2 graded Hilbert space which is a representation of the Lie superalgebra 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 algebra representation of a Lie superalgebra. 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,
- L[a|ψ>)]=(L[a])|ψ>+(-1)Laa(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)LaaL
The attractive feature of this approach is that there is no need to introduce (mysterious) Grassmann numbers.
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