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      

On (2)-relative cohomology of the Lie algebra of vector fields and differential operators

Abstract

Let Vect(?) be the Lie algebra of smooth vector fields on ?. The space of symbols Pol(T*?) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(?)-module that becomes trivial once the action is restricted to (2) ? Vect(?). The deformations of Pol(T*?), which become trivial once the action is restricted to (2) and such that the Vect(?)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D _λ,μ = Homdiff(F _λ, F _μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).     

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.