Geodesic
Shapovalov determinant for loop superalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 378-397 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Lie superalgebra, Shapovalov determinant. 
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A. V. Lebedev; D. A. Leites. Shapovalov determinant for loop superalgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 378-397. http://geodesic.mathdoc.fr/item/TMF_2008_156_3_a4/

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