Deformation of $\mathfrak {aff}(1)$-modules of pseudo-differential operators and symbols

Abstract

We consider the action of the Lie algebra $\mathfrak {aff}(1)$, by the Lie derivative on the space of symbols $\mathcal S_{\delta }=\bigoplus _{k=0}^{+\infty }\mathcal {F}_{\delta -k}$ and of pseudo-differential operators $\Psi \mathcal {D}\mathcal {O}$. We investigate the first and the second cohomology space associated with the embedding of the affine Lie algebra $\mathfrak {aff}(1)$ on the line $\mathbb {R}$ in the modules $\mathrm {M}$ where $\mathrm {M}=\mathcal {F}_{\lambda }$, $\mathrm {D}_{\lambda ,\mu }$, $\mathcal {S}_\delta$, $\mathcal {P}$, $\Psi \mathcal {D}\mathcal {O}$. We study the deformations of the structure of the $\mathfrak {aff}(1)$-modules $\mathcal {S}_\delta$, $\mathcal {P}$ and $\Psi \mathcal {D}\mathcal {O}$. We prove that any formal deformation is equivalent to its infinitesimal part. We complete our study by giving a example of deformation. Following the work of Guha we study some applications in mathematical physics.