Shapovalov determinant for loop superalgebras
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arXiv
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| Format: | Preprint |
| Published: |
2013
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| _version_ | 1866914948423090176 |
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| author | Lebedev, Alexei Leites, Dimitry |
| author_facet | Lebedev, Alexei Leites, Dimitry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1309_2542 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Shapovalov determinant for loop superalgebras Lebedev, Alexei Leites, Dimitry Representation Theory 81T30 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. |
| title | Shapovalov determinant for loop superalgebras |
| topic | Representation Theory 81T30 |
| url | https://arxiv.org/abs/1309.2542 |