The Shapovalov Determinant for the Poisson Superalgebras

DOI

10.2991/jnmp.2001.8.2.6How to use a DOI?

Abstract

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 C2: these are the Lie superalgebra kL (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 Kac­Moody version of sh(0|2k). Using C2 we compute N. Shapovalov determinant for kL (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).

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Pavel Grozman
AU  - Dimitry Leites
PY  - 2001
DA  - 2001/05/01
TI  - The Shapovalov Determinant for the Poisson Superalgebras
JO  - Journal of Nonlinear Mathematical Physics
SP  - 220
EP  - 228
VL  - 8
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2001.8.2.6
DO  - 10.2991/jnmp.2001.8.2.6
ID  - Grozman2001
ER  -