Shapovalov determinant for loop superalgebras

TL;DR

This paper explores the structure of Casimir elements in loop superalgebras derived from finite-dimensional simple Lie superalgebras, providing explicit formulas and extending known results to new classes.

Contribution

It introduces an explicit Wick normal form for the quadratic Casimir operator of Kac-Moody superalgebras associated with g, and computes the cubic Casimir for superalgebras with odd invariant forms.

Findings

Explicit Wick normal form of the quadratic Casimir operator.

Recovery of Casimir elements for loop superalgebras from finite-dimensional cases.

Extension of Casimir element computations to Poisson Lie 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.