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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 $\mathrm{STr}(\pi (L_a ))$ ?

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A representation of a superalgebra is typically a map to $\mathfrak{gl}(m|n)$. In other words, it is an action on a $\mathbb{Z}_2$-graded vector space. Then the supertrace comes from the usual one on $\mathfrak{gl}(m|n)$. Of course, you can take $n=0$, and consider a representation on an ordinary vector space. Then the supertrace is just the trace.

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