Representation of a Lie superalgebra

Definition

Let $\mathfrak g$ be a Lie superalgebra and let $V$ be a super vector space. A representation of $\mathfrak g$ on $V$ is an even Lie-superalgebra morphism

where $\mathfrak{gl}(V)=\underline{\operatorname{End}}(V)$ is the super internal endomorphism space with its supercommutator bracket.

Equivalently, homogeneous $x\in\mathfrak g$ acts by an endomorphism of parity $|x|$, and

The adjoint representation is the action on $\mathfrak g$ given by $\operatorname{ad}_x(y)=[x,y]$. The super Jacobi identity is exactly the identity that makes this a representation.