In this article, we provide a specific 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.

Weitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so-called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non-symmetric holonomy groups. We explicitly construct a basis of the space of Weitzenböck formulas. This classification allows us to find customized Weitzenböck formulas for applications such as eigenvalue estimates or Betti number estimates.