Unitary representation of a star Lie superalgebra - Definition and Overview

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.