The Shapovalov Determinant for the Poisson Superalgebras

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

A note on Bernoulli polynomials and solitons

Abstract

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula expressing the moments of the one-soliton function sech²(x-t) in terms of the Bernoulli polynomials Bn(x). We also provide an alternative short proof to the Grosset-Veselov formula connecting the one-soliton to the Bernoulli numbers (D?=?d/dx) published recently in this journal.     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.