Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$

Abstract

Over the $(1,n)$-dimensional real supercircle, we consider the $\mathcal{K}(n)$-modules of linear differential operators, ${\mathfrak{D}}_{\lambda ,\mu }^n$, acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules for $n=1$. We also prove that ${\mathfrak{D}}_{\lambda ,\mu }^n$ and ${\mathfrak{D}}_{\rho ,\nu }^n$ are isomorphic for $\rho =\frac{2-n}{2}-\mu$ and $\nu =\frac{2-n}{2}-\lambda$. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of Differential Operators on the Real Line, Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 13--18, 2001.]

Cited by in corpus (1)

• Conformally equivariant quantization and symbol maps associated with $n$-ary differential operators on weighted densities