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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2001
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0104287 |
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| _version_ | 1866914950209863680 |
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| author | Grozman, Pavel Leites, Dimitry |
| author_facet | Grozman, Pavel Leites, Dimitry |
| contents | Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0104287 |
| institution | arXiv |
| publishDate | 2001 |
| record_format | arxiv |
| spellingShingle | The Shapovalov determinant for the Poisson superalgebras Grozman, Pavel Leites, Dimitry Quantum Algebra Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k). |
| title | The Shapovalov determinant for the Poisson superalgebras |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/math/0104287 |