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 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).