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The Shapovalov Determinant for the Poisson Superalgebras
Abstract:Abstract

Among simple ?-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 /></span> of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra <span class= /></span> of special Hamiltonian fields in 2<i>k</i> odd indeterminates, and the Kac–Moody version of <span class= /></span>. Using <i>C</i> <sub>2</sub> we compute N. Shapovalov determinant for <span class= /></span> and <span class= /></span>, and for the Poisson superalgebras <span class= /></span> associated with <span class= /></span>. A. Shapovalov described irreducible finite dimensional representations of <span class= /></span> and <span class= /></span>; we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over <span class= /></span> and <span class= /></span></td> </tr> <tr> <td align=
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