Representation of a Lie superalgebra
# Representation of a Lie superalgebra

In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a $Z_2-graded$ 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

    $(c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X$
    $A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y$
    $(-1)^{A\cdot X}=(-1)^{A}(-1)^{X}$
    $[A,B]\cdot X=A\cdot (B\cdot X)-(-1)^{AB}B\cdot (A\cdot X).$

Equivalently, a representation of L is a $Z_2-graded$ 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 involutive antilinear map $^*$ such that * respects the grading and

    $[a,b]^*=[b^*,a^*].$

A unitary representation of such a Lie algebra is a $Z_2$ 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 *-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 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,

    $L[a|ψ>)]=(L[a])|ψ>+(-1)^{La}a(L[|ψ>]).$

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

    $L[a]=La-(-1)^{La}aL.$

This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers.

