In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{ g}$-invariants of the tensor algebra $T(\mathfrak{ g})$, the supersymmetric algebra $S(\mathfrak{ g})$, and the universal enveloping algebra $\mathrm{ U}(\mathfrak{ g})$ of a classical Lie superalgebra $\mathfrak{ g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak{ g})^{ \mathfrak{ g}}$ of the projection from $T(\mathfrak{ g})$ to $\mathrm{ U}(\mathfrak{ g})$ is surjective, which enables us to determine the generators of the center $\mathcal{ Z}(\mathfrak{ g})$ except for $\mathfrak{ g}=\mathfrak{ osp}_{ 2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal{ Z}(\mathfrak{ p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams.
Categories: math.RT