Shapovalov determinant for loop superalgebras

Alexei Lebedev; Dimitry Leites

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arXiv · 1309.2542

Shapovalov determinant for loop superalgebras

Alexei Lebedev·Dimitry Leites

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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.

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BibTeX RIS

Alexei Lebedev, Dimitry Leites. 2013-09-10. Shapovalov determinant for loop superalgebras. https://arxiv.org/abs/1309.2542

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