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Shapovalov determinant for loop superalgebras

Alexei Lebedev, Dimitry Leites

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.

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.

Paper Structure

This paper contains 21 sections, 68 equations.

Table of Contents

  1. Introduction
  2. Motivations: physics
  3. Means: mathematics
  4. Cases where an even quadratic Casimir exists
  5. Cases where no even quadratic Casimirs exist
  6. Our results and open problems
  7. Background: Lie superalgebras
  8. The Poisson superalgebra
  9. The integral
  10. Cartan subalgebras, maximal tori, roots and coroots
  11. The Shapovalov determinant
  12. Casimir operators on Lie superalgebras: the case of an even invariant form
  13. Twisted Kac--Moody (super)algebras
  14. Casimir operators on Lie superalgebras: the case of an odd form
  15. Kac--Moody-type Lie superalgebras based on queer Lie superalgebras
  16. ...and 6 more sections

Theorems & Definitions (4)

  • proof
  • proof
  • proof
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