Abstract:
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.