Shapovalov determinant for loop superalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 3, pp. 378-397
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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.
@article{TMF_2008_156_3_a4,
author = {A. V. Lebedev and D. A. Leites},
title = {Shapovalov determinant for loop superalgebras},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {378--397},
publisher = {mathdoc},
volume = {156},
number = {3},
year = {2008},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2008_156_3_a4/}
}
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/