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).     

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      

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.     

Fractional q-Calculus on a time scale

Abstract

The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale , where to ∈ ? and 0 < q < 1. By use of time scale calculus notation, we find the proof of many results more straight forward. We shall develop some properties of fractional q-calculus, we shall develop some properties of a q-Laplace transform, and then we shall employ the q-Laplace transform to solve fractional q-difference equations.