homomorphism of Lie superalgebras

In the book Shun-Jen Cheng, Weiqiang Wang Dualities and Representations of Lie Superalgebrasm. One founds the following definition(Definition 1.3):Let $\mathfrak{g}$ and $\mathfrak{g'}$ be Lie superalgebras. A homomorphism of Lie superalgebras is an even linear map $f: \mathfrak{g} \rightarrow \mathfrak{g'}$ satisfying$$f([a,b])=[f(a),f(b)],~ a, b \in \mathfrak{g}. ~~~~(*)$$Here is my question:Must a homomorphism of Lie superalgebras be even? Assume $\mathfrak{g}$ is a Lie superalgebra, $A$ is a trivial $\mathfrak{g}$-supermodule. Then $A$ can...Read more

rt.representation theory - Definition of the supertrace in superalgebra representations

Let us consider a matrix superalgebra $A$ with generators satisfying $[L_a,L_b]=i L_c f^c{}_{ab}.$ The generators are matrices on which supertrace is defined bu the usual trace on the bosonic part minus the usual trace on the fermionic part. But then let $\pi : A \longrightarrow gl(V)$ be a representation of the superalgebra. We know that an invariant form in this representation is given by $B_{ab}=\mathrm{STr}(\pi (L_a ) \pi (L_b))$, but I don't understand how the supertrace is defined in the representation. In other words, what is $\mathrm{ST...Read more