Shapovalov determinant for loop superalgebras

Abstract

Let a given finite dimensional simple Lie superalgebra g possess an even invariant non-degenerate supersymmetric bilinear form. We show how to recover the quadratic Casimir element for the Kac-Moody superalgebra related to the loop superalgebra with values in g from the quadratic Casimir element for g. Our main tool here is an explicit Wick normal form of the even quadratic Casimir operator for the Kac--Moody superalgebra associated with g; this Wick normal form is of independent interest. If g possesses an odd invariant supersymmetric bilinear form we compute the cubic Casimir element. In addition to the cases of Lie superalgebras g(A) with Cartan matrix A for which the answer was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac--Moody superalgebra.