Shapovalov determinant for loop superalgebras

For the Kac–Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra $\mathfrak g$, we show what its quadratic Casimir element is equal to if the Casimir element for $\mathfrak g$ is known (if $\mathfrak g$ has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac–Moody superalgebra associated with $\mathfrak g$; this Wick normal form is independently interesting. If $\mathfrak g$ has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras $\mathfrak g=\mathfrak g(A)$ with a Cartan matrix $A$ for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra $\mathfrak{poi}(0\mid n)$ and the related Kac–Moody superalgebra.