Mabrouk Ben Ammar

COHOMOLOGY OF THE LIE SUPERALGEBRA OF CONTACT VECTOR FIELDS ON 𝕂1|1 AND DEFORMATIONS OF THE SUPERSPACE OF SYMBOLS

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but (1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal (1|2)-trivial deformations of the -module structure on the superspaces of symbols of differential operators. We prove that any generic formal (1|2)-trivial deformation of this -module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On (2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to (N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.

The binary invariant differential operators on weighted densities on the superspace R1∣n and cohomology

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.

Deformation of vect(1)-modules of symbols

We consider the action of the Lie algebra of polynomial vector fields,

The Poincare-Dulac theorem for nonlinear representations of nilpotent lie algebras

\mathfrak{vect}(1)

Differential operators on the weighted densities on the supercircle S1|1

, by the Lie derivative on the space of symbols

Deformation of -modules of symbols

\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}

Deformation of vect(1)vect(1)-modules of symbols

. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space

Deformation of Vect(

\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{\nu,\mu})

\mathbb{R})

where

-Modules of Symbols

{\cal D}_{\nu,\mu}